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First Year Math Trigonometric Identities Prove the following identities:3. \frac{\sin 2 \alpha}{1+\cos 2 \alpha}=\tan \alpha


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Prove the following identities:3. \frac{\sin 2 \alpha}{1+\cos 2 \alpha}=\tan \alpha

1. Prove thatv) \cos \left(270^{\circ}+\theta\right)=\sin \theta
1. Prove thatv)   \cos \left(270^{\circ}+\theta\right)=\sin \theta

1. Prove thatv) \cos \left(270^{\circ}+\theta\right)=\sin \theta

3. Prove that:ii) \cos \left(\alpha+45^{\circ}\right)=\frac{1}{\sqrt{2}}(\cos \alpha-\sin \alpha)
3. Prove that:ii)   \cos \left(\alpha+45^{\circ}\right)=\frac{1}{\sqrt{2}}(\cos \alpha-\sin \alpha)

3. Prove that:ii) \cos \left(\alpha+45^{\circ}\right)=\frac{1}{\sqrt{2}}(\cos \alpha-\sin \alpha)

2. Express the following sums or differences as products:ii) \sin 8 \theta-\sin 4 \theta
2. Express the following sums or differences as products:ii)  \sin 8 \theta-\sin 4 \theta

2. Express the following sums or differences as products:ii) \sin 8 \theta-\sin 4 \theta

Example 5: If \alpha \beta \gamma are the angles of \triangle A B C prove that:i) \tan \alpha+\tan \beta+\tan \gamma=\tan \alpha \tan \beta \tan \gamma
Example 5: If  \alpha \beta \gamma  are the angles of  \triangle A B C  prove that:i)  \tan \alpha+\tan \beta+\tan \gamma=\tan \alpha \tan \beta \tan \gamma

Example 5: If \alpha \beta \gamma are the angles of \triangle A B C prove that:i) \tan \alpha+\tan \beta+\tan \gamma=\tan \alpha \tan \beta \tan \gamma

Prove the following identities:3. \frac{\sin 2 \alpha}{1+\cos 2 \alpha}=\tan \alpha
Prove the following identities:3.  \frac{\sin 2 \alpha}{1+\cos 2 \alpha}=\tan \alpha
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Prove the following identities:3. \frac{\sin 2 \alpha}{1+\cos 2 \alpha}=\tan \alpha

9. If \sin \alpha=\frac{4}{5}< and \sin \beta=\frac{12}{13}< where \frac{\pi}{2} \alpha \pi and \frac{\pi}{2} \beta \pi . Findii) \cos (\alpha+\beta) In which rants do the terminal sides of the angles of measures (\alpha+\beta) and (\alpha-\beta) lie?
9. If  \sin \alpha=\frac{4}{5}<  and  \sin \beta=\frac{12}{13}<  where  \frac{\pi}{2}  \alpha  \pi  and  \frac{\pi}{2}  \beta  \pi . Findii)   \cos (\alpha+\beta) In which rants do the terminal sides of the angles of measures  (\alpha+\beta)  and  (\alpha-\beta)  lie?

9. If \sin \alpha=\frac{4}{5}< and \sin \beta=\frac{12}{13}< where \frac{\pi}{2} \alpha \pi and \frac{\pi}{2} \beta \pi . Findii) \cos (\alpha+\beta) In which rants do the terminal sides of the angles of measures (\alpha+\beta) and (\alpha-\beta) lie?

5. If \alpha \beta \gamma are the angles of a triangle A B C then prove thatii) \cos \left(\frac{\alpha+\beta}{2}\right)=\sin \frac{\gamma}{2}
5. If  \alpha \beta \gamma  are the angles of a triangle  A B C  then prove thatii)  \cos \left(\frac{\alpha+\beta}{2}\right)=\sin \frac{\gamma}{2}

5. If \alpha \beta \gamma are the angles of a triangle A B C then prove thatii) \cos \left(\frac{\alpha+\beta}{2}\right)=\sin \frac{\gamma}{2}

1. Without using the tables find the values of:\[\text { ii) } \cot \left(-855^{\prime}\right)\]
1. Without using the tables find the values of:\[\text { ii) } \cot \left(-855^{\prime}\right)\]

1. Without using the tables find the values of:\[\text { ii) } \cot \left(-855^{\prime}\right)\]

11. Prove that \frac{\cos 8^{\circ}}{\cos 8^{\circ}} \frac{\sin 8^{\circ}}{\sin 8^{\circ}} \tan 37
11. Prove that  \frac{\cos 8^{\circ}}{\cos 8^{\circ}} \frac{\sin 8^{\circ}}{\sin 8^{\circ}}  \tan 37

11. Prove that \frac{\cos 8^{\circ}}{\cos 8^{\circ}} \frac{\sin 8^{\circ}}{\sin 8^{\circ}} \tan 37

Example 2: Without using the tables write down the values of:i) \cos 315^{\circ}
Example 2: Without using the tables write down the values of:i)  \cos 315^{\circ}

Example 2: Without using the tables write down the values of:i) \cos 315^{\circ}

9. If \sin \alpha=\frac{4}{5}< and \sin \beta=\frac{12}{13}< where \frac{\pi}{2} \alpha \pi and \frac{\pi}{2} \beta \pi . Findiv) \sin (\alpha-\beta) In which rants do the terminal sides of the angles of measures (\alpha+\beta) and (\alpha-\beta) lie?
9. If  \sin \alpha=\frac{4}{5}<  and  \sin \beta=\frac{12}{13}<  where  \frac{\pi}{2}  \alpha  \pi  and  \frac{\pi}{2}  \beta  \pi . Findiv)  \sin (\alpha-\beta) In which rants do the terminal sides of the angles of measures  (\alpha+\beta)  and  (\alpha-\beta)  lie?

9. If \sin \alpha=\frac{4}{5}< and \sin \beta=\frac{12}{13}< where \frac{\pi}{2} \alpha \pi and \frac{\pi}{2} \beta \pi . Findiv) \sin (\alpha-\beta) In which rants do the terminal sides of the angles of measures (\alpha+\beta) and (\alpha-\beta) lie?

1. Express the following products as sums or differences:vi) \cos \left(2 x+30^{\circ}\right) \cos \left(2 x-30^{\circ}\right)
1. Express the following products as sums or differences:vi)  \cos \left(2 x+30^{\circ}\right) \cos \left(2 x-30^{\circ}\right)

1. Express the following products as sums or differences:vi) \cos \left(2 x+30^{\circ}\right) \cos \left(2 x-30^{\circ}\right)

Q.1 \cos (-\alpha)= (a) \cos \alpha (b) -\sin a (c) -\cos \alpha (d) \sin \alpha
Q.1  \cos (-\alpha)= (a)  \cos \alpha (b)  -\sin a (c)  -\cos \alpha (d)  \sin \alpha

Q.1 \cos (-\alpha)= (a) \cos \alpha (b) -\sin a (c) -\cos \alpha (d) \sin \alpha

4. Prove that:i) \cos 20^{\circ}+\cos 100^{\circ}+\cos 140^{\circ}=0
4. Prove that:i)   \cos 20^{\circ}+\cos 100^{\circ}+\cos 140^{\circ}=0

4. Prove that:i) \cos 20^{\circ}+\cos 100^{\circ}+\cos 140^{\circ}=0

13. If \alpha+\beta+\gamma=180^{\circ} show that\[\cot \alpha \cot \beta+\cot \beta \cot \gamma+\cot \gamma \cot \alpha=1\]
13. If  \alpha+\beta+\gamma=180^{\circ}  show that\[\cot \alpha \cot \beta+\cot \beta \cot \gamma+\cot \gamma \cot \alpha=1\]

13. If \alpha+\beta+\gamma=180^{\circ} show that\[\cot \alpha \cot \beta+\cot \beta \cot \gamma+\cot \gamma \cot \alpha=1\]

4. Prove that:iii) \frac{\sin \theta+\sin 3 \theta+\sin 5 \theta+\sin 7 \theta}{\cos \theta+\cos 3 \theta+\cos 5 \theta+\cos 7 \theta}=\tan 4 \theta
4. Prove that:iii)  \frac{\sin \theta+\sin 3 \theta+\sin 5 \theta+\sin 7 \theta}{\cos \theta+\cos 3 \theta+\cos 5 \theta+\cos 7 \theta}=\tan 4 \theta

4. Prove that:iii) \frac{\sin \theta+\sin 3 \theta+\sin 5 \theta+\sin 7 \theta}{\cos \theta+\cos 3 \theta+\cos 5 \theta+\cos 7 \theta}=\tan 4 \theta

1. Prove thati) \sin \left(180^{\circ}+\theta\right)=-\sin \theta
1. Prove thati)   \sin \left(180^{\circ}+\theta\right)=-\sin \theta

1. Prove thati) \sin \left(180^{\circ}+\theta\right)=-\sin \theta

9. If \sin \alpha=\frac{4}{5}< and \sin \beta=\frac{12}{13}< where \frac{\pi}{2} \alpha \pi and \frac{\pi}{2} \beta \pi . Findi) \sin (\alpha+\beta) In which rants do the terminal sides of the angles of measures (\alpha+\beta) and (\alpha-\beta) lie?
9. If  \sin \alpha=\frac{4}{5}<  and  \sin \beta=\frac{12}{13}<  where  \frac{\pi}{2}  \alpha  \pi  and  \frac{\pi}{2}  \beta  \pi . Findi)  \sin (\alpha+\beta) In which rants do the terminal sides of the angles of measures  (\alpha+\beta)  and  (\alpha-\beta)  lie?

9. If \sin \alpha=\frac{4}{5}< and \sin \beta=\frac{12}{13}< where \frac{\pi}{2} \alpha \pi and \frac{\pi}{2} \beta \pi . Findi) \sin (\alpha+\beta) In which rants do the terminal sides of the angles of measures (\alpha+\beta) and (\alpha-\beta) lie?

8. If \sin \alpha=\frac{4}{5} and \cos \beta \frac{40}{41} where 0<\alpha<\frac{\pi}{2} and 0<\beta<\frac{\pi}{2} . Show that \sin (\alpha-\beta)=\frac{133}{205} .
8. If  \sin \alpha=\frac{4}{5}  and  \cos \beta  \frac{40}{41}  where  0<\alpha<\frac{\pi}{2}  and  0<\beta<\frac{\pi}{2} . Show that  \sin (\alpha-\beta)=\frac{133}{205} .
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8. If \sin \alpha=\frac{4}{5} and \cos \beta \frac{40}{41} where 0<\alpha<\frac{\pi}{2} and 0<\beta<\frac{\pi}{2} . Show that \sin (\alpha-\beta)=\frac{133}{205} .

Prove the following identities:7. \frac{\operatorname{coses} \theta+\operatorname{coses} 2 \theta}{\sec \theta}=\cot \frac{\theta}{2}
Prove the following identities:7.  \frac{\operatorname{coses} \theta+\operatorname{coses} 2 \theta}{\sec \theta}=\cot \frac{\theta}{2}
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Prove the following identities:7. \frac{\operatorname{coses} \theta+\operatorname{coses} 2 \theta}{\sec \theta}=\cot \frac{\theta}{2}

Example 2: Show thatii) \cos 2 \theta=\frac{1-\tan ^{2} \theta}{1+\tan ^{2} \theta}
Example 2: Show thatii)  \cos 2 \theta=\frac{1-\tan ^{2} \theta}{1+\tan ^{2} \theta}
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Example 2: Show thatii) \cos 2 \theta=\frac{1-\tan ^{2} \theta}{1+\tan ^{2} \theta}

Q.22 \sin 2 \theta equals(a) \sin \theta \cos \theta (b) 2 \sin \theta \cos \theta (c) 1-2 \sin ^{2} \theta (d) 2 \cdot \cos ^{2} \theta-1
Q.22  \sin 2 \theta  equals(a)  \sin \theta \cos \theta  (b)  2 \sin \theta \cos \theta  (c)  1-2 \sin ^{2} \theta (d)  2 \cdot \cos ^{2} \theta-1
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Q.22 \sin 2 \theta equals(a) \sin \theta \cos \theta (b) 2 \sin \theta \cos \theta (c) 1-2 \sin ^{2} \theta (d) 2 \cdot \cos ^{2} \theta-1

1. Find the values of \sin 2 \alpha \cos 2 \alpha and \tan 2 \alpha when:12. \frac{\tan \frac{\theta}{2}+\cot \frac{\theta}{2}}{\cot \frac{\theta}{2}-\tan \frac{\theta}{2}}=\sec \theta
1. Find the values of  \sin 2 \alpha \cos 2 \alpha  and  \tan 2 \alpha  when:12.  \frac{\tan \frac{\theta}{2}+\cot \frac{\theta}{2}}{\cot \frac{\theta}{2}-\tan \frac{\theta}{2}}=\sec \theta
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1. Find the values of \sin 2 \alpha \cos 2 \alpha and \tan 2 \alpha when:12. \frac{\tan \frac{\theta}{2}+\cot \frac{\theta}{2}}{\cot \frac{\theta}{2}-\tan \frac{\theta}{2}}=\sec \theta

Q.18 \cot \left(90^{\circ}-\alpha\right) equals :(a) =\tan \alpha (b) \tan \alpha (c) \cot \alpha (d) -\cot \alpha Lahore Board 2016
Q.18  \cot \left(90^{\circ}-\alpha\right)  equals :(a)  =\tan \alpha   (b)  \tan \alpha (c)  \cot \alpha (d)  -\cot \alpha Lahore Board 2016
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Q.18 \cot \left(90^{\circ}-\alpha\right) equals :(a) =\tan \alpha (b) \tan \alpha (c) \cot \alpha (d) -\cot \alpha Lahore Board 2016

15. Find the values of \sin \theta and \cos \theta without using table or calculator when \theta isi) 18^{\circ} Hence prove that: \cos 36^{\circ} \cos 72^{\circ} \cos 108^{\circ} \cos 144^{\circ}=\frac{1}{16}
15. Find the values of  \sin \theta  and  \cos \theta  without using table or calculator when  \theta  isi)   18^{\circ} Hence prove that:  \cos 36^{\circ} \cos 72^{\circ} \cos 108^{\circ} \cos 144^{\circ}=\frac{1}{16}
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15. Find the values of \sin \theta and \cos \theta without using table or calculator when \theta isi) 18^{\circ} Hence prove that: \cos 36^{\circ} \cos 72^{\circ} \cos 108^{\circ} \cos 144^{\circ}=\frac{1}{16}

Q.44 If \cot \theta<0 \cos \theta<0 then the terminal arm of the angle lies in the rant(a) 1(b) \| (c) III(d) IV
Q.44 If  \cot \theta<0 \cos \theta<0  then the terminal arm of the angle lies in the rant(a) 1(b)  \| (c) III(d) IV
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Q.44 If \cot \theta<0 \cos \theta<0 then the terminal arm of the angle lies in the rant(a) 1(b) \| (c) III(d) IV

14. Express the following in the form r \sin (\theta+\phi) or r \sin (\theta-\phi) where terminal sides of the angles of measures \theta and \phi are in the first rant:ii) 3 \sin \theta-4 \cos \theta
14. Express the following in the form  r \sin (\theta+\phi)  or  r \sin (\theta-\phi)  where terminal sides of the angles of measures  \theta  and  \phi  are in the first rant:ii)  3 \sin \theta-4 \cos \theta
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14. Express the following in the form r \sin (\theta+\phi) or r \sin (\theta-\phi) where terminal sides of the angles of measures \theta and \phi are in the first rant:ii) 3 \sin \theta-4 \cos \theta

Q.30 2 \sin \alpha \cos \beta= (a) \cos (\alpha+\beta)+\cos (\alpha-\beta) (b) \cos (\alpha+\beta)-\cos (\alpha-\beta) (c) \sin (\alpha+\beta)+\sin (\alpha-\beta) (d) \sin (\alpha+\beta)-\sin (\alpha-\beta)
Q.30  2 \sin \alpha \cos \beta= (a)  \cos (\alpha+\beta)+\cos (\alpha-\beta) (b)  \cos (\alpha+\beta)-\cos (\alpha-\beta) (c)  \sin (\alpha+\beta)+\sin (\alpha-\beta) (d)  \sin (\alpha+\beta)-\sin (\alpha-\beta)
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Q.30 2 \sin \alpha \cos \beta= (a) \cos (\alpha+\beta)+\cos (\alpha-\beta) (b) \cos (\alpha+\beta)-\cos (\alpha-\beta) (c) \sin (\alpha+\beta)+\sin (\alpha-\beta) (d) \sin (\alpha+\beta)-\sin (\alpha-\beta)

9. If \sin \alpha=\frac{4}{5}< and \sin \beta=\frac{12}{13}< where \frac{\pi}{2} \alpha \pi and \frac{\pi}{2} \beta \pi . Findvi) \tan (\alpha-\beta) .In which rants do the terminal sides of the angles of measures (\alpha+\beta) and (\alpha-\beta) lie?
9. If  \sin \alpha=\frac{4}{5}<  and  \sin \beta=\frac{12}{13}<  where  \frac{\pi}{2}  \alpha  \pi  and  \frac{\pi}{2}  \beta  \pi . Findvi)  \tan (\alpha-\beta) .In which rants do the terminal sides of the angles of measures  (\alpha+\beta)  and  (\alpha-\beta)  lie?
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9. If \sin \alpha=\frac{4}{5}< and \sin \beta=\frac{12}{13}< where \frac{\pi}{2} \alpha \pi and \frac{\pi}{2} \beta \pi . Findvi) \tan (\alpha-\beta) .In which rants do the terminal sides of the angles of measures (\alpha+\beta) and (\alpha-\beta) lie?

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