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First Year Math Inverse Trigonometric Functions Q.20 \operatorname{Tan}^{-1}\left(\frac{A-B}{1+A B}\right) is equal to:(a) \operatorname{Tan}^{-1} A+\operatorname{Tan}^{-1} B (b) \operatorname{Tan}^{-1} A-\operatorname{Tan}^{-1} B (c) \operato


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Class 9Class 10First YearSecond Year
Q.20 \operatorname{Tan}^{-1}\left(\frac{A-B}{1+A B}\right) is equal to:(a) \operatorname{Tan}^{-1} A+\operatorname{Tan}^{-1} B (b) \operatorname{Tan}^{-1} A-\operatorname{Tan}^{-1} B (c) \operatorname{Cot}^{-1} A+\operatorname{Cot}^{-1} B (d) \operatorname{Cot}^{-1} A-\operatorname{Cot}^{-1} B

Q.21 \operatorname{Tan}^{-1} \frac{1}{2}+\operatorname{Tan}^{-1} \frac{1}{3}= (a) \frac{\pi}{6} (b) \frac{\pi}{4} (c) \frac{\pi}{2} (d) \pi Gujrantwala Board; 2008
Q.21  \operatorname{Tan}^{-1} \frac{1}{2}+\operatorname{Tan}^{-1} \frac{1}{3}= (a)  \frac{\pi}{6} (b)  \frac{\pi}{4} (c)  \frac{\pi}{2} (d)  \pi Gujrantwala Board; 2008

Q.21 \operatorname{Tan}^{-1} \frac{1}{2}+\operatorname{Tan}^{-1} \frac{1}{3}= (a) \frac{\pi}{6} (b) \frac{\pi}{4} (c) \frac{\pi}{2} (d) \pi Gujrantwala Board; 2008

16. \tan 28 \cot \theta=3
16.  \tan 28 \cot \theta=3

16. \tan 28 \cot \theta=3

11. \operatorname{Tan}^{-1} \frac{1}{13}+\operatorname{Tan}^{-1} \frac{1}{4}=\operatorname{Tan}^{-1} \frac{1}{3}
11.  \operatorname{Tan}^{-1} \frac{1}{13}+\operatorname{Tan}^{-1} \frac{1}{4}=\operatorname{Tan}^{-1} \frac{1}{3}

11. \operatorname{Tan}^{-1} \frac{1}{13}+\operatorname{Tan}^{-1} \frac{1}{4}=\operatorname{Tan}^{-1} \frac{1}{3}

8. Find \sin \left(\sin ^{-1} \frac{3}{5}+\sin ^{-1} \frac{4}{5}\right) Prove that
8. Find  \sin \left(\sin ^{-1} \frac{3}{5}+\sin ^{-1} \frac{4}{5}\right)  Prove that

8. Find \sin \left(\sin ^{-1} \frac{3}{5}+\sin ^{-1} \frac{4}{5}\right) Prove that

\[\sin \left(\cos ^{-1} \theta+\sin ^{-1} \theta\right)=\text { ? }\]A. 4B. 3C. 2D. 1
\[\sin \left(\cos ^{-1} \theta+\sin ^{-1} \theta\right)=\text { ? }\]A. 4B. 3C. 2D. 1

\[\sin \left(\cos ^{-1} \theta+\sin ^{-1} \theta\right)=\text { ? }\]A. 4B. 3C. 2D. 1

Example 3. Solve 2 \sin ^{2} \theta+2 \sqrt{2} \sin \theta-3=0 .
Example 3. Solve  2 \sin ^{2} \theta+2 \sqrt{2} \sin \theta-3=0 .

Example 3. Solve 2 \sin ^{2} \theta+2 \sqrt{2} \sin \theta-3=0 .

Q.28 y=\cos x is one to one in interval:(a) \left[0 \frac{3 \pi}{2}\right] (b) [02 \pi] (c) [0 \infty] (d) [0 \pi] Lahore Board 2016
Q.28  y=\cos x  is one to one in interval:(a)  \left[0 \frac{3 \pi}{2}\right] (b)  [02 \pi] (c)  [0 \infty] (d)  [0 \pi] Lahore Board 2016

Q.28 y=\cos x is one to one in interval:(a) \left[0 \frac{3 \pi}{2}\right] (b) [02 \pi] (c) [0 \infty] (d) [0 \pi] Lahore Board 2016

\tan \left(\cot ^{-1} x\right) is equal toA. \tan \mathrm{x} B. \frac{\pi}{2}-x C. \cot \left(\tan ^{-1} \mathrm{x}\right) D. None of these
 \tan \left(\cot ^{-1} x\right)  is equal toA.  \tan \mathrm{x} B.  \frac{\pi}{2}-x C.  \cot \left(\tan ^{-1} \mathrm{x}\right) D. None of these

\tan \left(\cot ^{-1} x\right) is equal toA. \tan \mathrm{x} B. \frac{\pi}{2}-x C. \cot \left(\tan ^{-1} \mathrm{x}\right) D. None of these

Q.12 \cos ^{-1}(-x) equals(a) -\cos ^{-1} x (b) \cos ^{-1} x (c) \pi-\cos ^{-1} x Lahore Board 2015; Gujranwala Board 2008
Q.12  \cos ^{-1}(-x)  equals(a)  -\cos ^{-1} x (b)  \cos ^{-1} x (c)  \pi-\cos ^{-1} x Lahore Board 2015; Gujranwala Board 2008

Q.12 \cos ^{-1}(-x) equals(a) -\cos ^{-1} x (b) \cos ^{-1} x (c) \pi-\cos ^{-1} x Lahore Board 2015; Gujranwala Board 2008

9. \operatorname{Cos}^{-1} x=\operatorname{Tan}^{-1} \frac{\sqrt{1-x^{2}}}{x} where 0<x \leq 1 Show that:
9.  \operatorname{Cos}^{-1} x=\operatorname{Tan}^{-1} \frac{\sqrt{1-x^{2}}}{x}  where  0<x \leq 1  Show that:

9. \operatorname{Cos}^{-1} x=\operatorname{Tan}^{-1} \frac{\sqrt{1-x^{2}}}{x} where 0<x \leq 1 Show that:

Q.5 The value of \sin \left(\operatorname{Cos}^{-1} \frac{\sqrt{3}}{2}\right) is(a) \frac{1}{2} (b) \frac{1}{\sqrt{2}} (c) \frac{\sqrt{3}}{2} (d) none of theseMultan Board 2004; Lahore Boărd 2011
Q.5 The value of  \sin \left(\operatorname{Cos}^{-1} \frac{\sqrt{3}}{2}\right)  is(a)  \frac{1}{2} (b)  \frac{1}{\sqrt{2}} (c)  \frac{\sqrt{3}}{2} (d) none of theseMultan Board 2004; Lahore Boărd 2011

Q.5 The value of \sin \left(\operatorname{Cos}^{-1} \frac{\sqrt{3}}{2}\right) is(a) \frac{1}{2} (b) \frac{1}{\sqrt{2}} (c) \frac{\sqrt{3}}{2} (d) none of theseMultan Board 2004; Lahore Boărd 2011

Q.16 \operatorname{Sin}^{-1}\left(A \sqrt{1-B^{2}}+B \sqrt{1-A^{2}}\right)= (a) \operatorname{Sin}^{-1} A+\operatorname{Sin}^{-1} B (b) \operatorname{Sin}^{-1} A-\operatorname{Sin}^{-1} B (c) \operatorname{Cos}^{-1} A+\operatorname{Cos}^{-1} B (d) \operatorname{Cos}^{1} A-\operatorname{Cos}^{-1} B
Q.16  \operatorname{Sin}^{-1}\left(A \sqrt{1-B^{2}}+B \sqrt{1-A^{2}}\right)= (a)  \operatorname{Sin}^{-1} A+\operatorname{Sin}^{-1} B (b)  \operatorname{Sin}^{-1} A-\operatorname{Sin}^{-1} B (c)  \operatorname{Cos}^{-1} A+\operatorname{Cos}^{-1} B (d)  \operatorname{Cos}^{1} A-\operatorname{Cos}^{-1} B

Q.16 \operatorname{Sin}^{-1}\left(A \sqrt{1-B^{2}}+B \sqrt{1-A^{2}}\right)= (a) \operatorname{Sin}^{-1} A+\operatorname{Sin}^{-1} B (b) \operatorname{Sin}^{-1} A-\operatorname{Sin}^{-1} B (c) \operatorname{Cos}^{-1} A+\operatorname{Cos}^{-1} B (d) \operatorname{Cos}^{1} A-\operatorname{Cos}^{-1} B

\sin ^{-1} x+\sin ^{-1} y=? A. \sin ^{-1}\left(x \sqrt{1-x^{2}}+y \sqrt{1+x^{2}}\right) B. \sin ^{-1}\left(x \sqrt{1-x^{2}}-y \sqrt{1-y^{2}}\right) C. \sin ^{-1}\left(x \sqrt{1+y^{2}}+y \sqrt{1-x^{2}}\right) D. \sin ^{-1}\left(x \sqrt{1-y^{2}}-y \sqrt{1-x^{2}}\right)
 \sin ^{-1} x+\sin ^{-1} y=? A.   \sin ^{-1}\left(x \sqrt{1-x^{2}}+y \sqrt{1+x^{2}}\right) B.  \sin ^{-1}\left(x \sqrt{1-x^{2}}-y \sqrt{1-y^{2}}\right) C.  \sin ^{-1}\left(x \sqrt{1+y^{2}}+y \sqrt{1-x^{2}}\right) D.  \sin ^{-1}\left(x \sqrt{1-y^{2}}-y \sqrt{1-x^{2}}\right)

\sin ^{-1} x+\sin ^{-1} y=? A. \sin ^{-1}\left(x \sqrt{1-x^{2}}+y \sqrt{1+x^{2}}\right) B. \sin ^{-1}\left(x \sqrt{1-x^{2}}-y \sqrt{1-y^{2}}\right) C. \sin ^{-1}\left(x \sqrt{1+y^{2}}+y \sqrt{1-x^{2}}\right) D. \sin ^{-1}\left(x \sqrt{1-y^{2}}-y \sqrt{1-x^{2}}\right)

Q.4 \operatorname{Tan}^{-1}(-\sqrt{3})= (a) \frac{2 \pi}{3} (b) -\frac{2 \pi}{3} (c) -\frac{\pi}{6} (d) -\frac{\pi}{3} Lahore Board 2010
Q.4  \operatorname{Tan}^{-1}(-\sqrt{3})= (a)  \frac{2 \pi}{3} (b)  -\frac{2 \pi}{3} (c)  -\frac{\pi}{6} (d)  -\frac{\pi}{3} Lahore Board 2010

Q.4 \operatorname{Tan}^{-1}(-\sqrt{3})= (a) \frac{2 \pi}{3} (b) -\frac{2 \pi}{3} (c) -\frac{\pi}{6} (d) -\frac{\pi}{3} Lahore Board 2010

What is the range of \operatorname{Sec} \theta A. \mathrm{Q}-\{\mathrm{x} \mid-1<\mathrm{x}<1\} B. \mathrm{Z}-\{\mathrm{x} \mid-1<\mathrm{x}<1\} C. \mathrm{N}-\{\mathrm{x} \mid-1<\mathrm{x}<1\} D. \mathrm{R}-\{\mathrm{x} \mid-1<\mathrm{x}<1\}
What is the range of  \operatorname{Sec} \theta A.  \mathrm{Q}-\{\mathrm{x} \mid-1<\mathrm{x}<1\} B.  \mathrm{Z}-\{\mathrm{x} \mid-1<\mathrm{x}<1\} C.  \mathrm{N}-\{\mathrm{x} \mid-1<\mathrm{x}<1\} D.  \mathrm{R}-\{\mathrm{x} \mid-1<\mathrm{x}<1\}

What is the range of \operatorname{Sec} \theta A. \mathrm{Q}-\{\mathrm{x} \mid-1<\mathrm{x}<1\} B. \mathrm{Z}-\{\mathrm{x} \mid-1<\mathrm{x}<1\} C. \mathrm{N}-\{\mathrm{x} \mid-1<\mathrm{x}<1\} D. \mathrm{R}-\{\mathrm{x} \mid-1<\mathrm{x}<1\}

Q.6 \cos \left(\tan ^{-1} \sqrt{3}\right) is equal to :(a) \frac{1}{2} (b) -\frac{1}{2} (c) \frac{\sqrt{3}}{2} (d) -\frac{\sqrt{3}}{2}
Q.6  \cos \left(\tan ^{-1} \sqrt{3}\right)  is equal to :(a)  \frac{1}{2} (b)  -\frac{1}{2} (c)  \frac{\sqrt{3}}{2} (d)  -\frac{\sqrt{3}}{2}

Q.6 \cos \left(\tan ^{-1} \sqrt{3}\right) is equal to :(a) \frac{1}{2} (b) -\frac{1}{2} (c) \frac{\sqrt{3}}{2} (d) -\frac{\sqrt{3}}{2}

Q.8 If \sin ^{-1} 0=\alpha then value of \alpha is :(a) \frac{\pi}{2} (b) \frac{\pi}{3} (c) \frac{\pi}{4} (d) 0
Q.8 If  \sin ^{-1} 0=\alpha  then value of  \alpha  is :(a)  \frac{\pi}{2} (b)  \frac{\pi}{3} (c)  \frac{\pi}{4} (d) 0

Q.8 If \sin ^{-1} 0=\alpha then value of \alpha is :(a) \frac{\pi}{2} (b) \frac{\pi}{3} (c) \frac{\pi}{4} (d) 0

Q.30 \sin ^{-1} x is equal to(a) \frac{\pi}{2}+\sin ^{-1} x (b) \frac{\pi}{2}-\sin ^{-1} x (c) \frac{\pi}{2}+\cos ^{-1} x (d) \frac{\pi}{2}-\cos ^{-1} x Sahlwal Board 2016
Q.30  \sin ^{-1} x  is equal to(a)  \frac{\pi}{2}+\sin ^{-1} x (b)  \frac{\pi}{2}-\sin ^{-1} x (c)  \frac{\pi}{2}+\cos ^{-1} x (d)  \frac{\pi}{2}-\cos ^{-1} x Sahlwal Board 2016

Q.30 \sin ^{-1} x is equal to(a) \frac{\pi}{2}+\sin ^{-1} x (b) \frac{\pi}{2}-\sin ^{-1} x (c) \frac{\pi}{2}+\cos ^{-1} x (d) \frac{\pi}{2}-\cos ^{-1} x Sahlwal Board 2016

Q.20 \operatorname{Tan}^{-1}\left(\frac{A-B}{1+A B}\right) is equal to:(a) \operatorname{Tan}^{-1} A+\operatorname{Tan}^{-1} B (b) \operatorname{Tan}^{-1} A-\operatorname{Tan}^{-1} B (c) \operatorname{Cot}^{-1} A+\operatorname{Cot}^{-1} B (d) \operatorname{Cot}^{-1} A-\operatorname{Cot}^{-1} B
Q.20  \operatorname{Tan}^{-1}\left(\frac{A-B}{1+A B}\right)  is equal to:(a)  \operatorname{Tan}^{-1} A+\operatorname{Tan}^{-1} B (b)  \operatorname{Tan}^{-1} A-\operatorname{Tan}^{-1} B (c)  \operatorname{Cot}^{-1} A+\operatorname{Cot}^{-1} B (d)  \operatorname{Cot}^{-1} A-\operatorname{Cot}^{-1} B
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Q.20 \operatorname{Tan}^{-1}\left(\frac{A-B}{1+A B}\right) is equal to:(a) \operatorname{Tan}^{-1} A+\operatorname{Tan}^{-1} B (b) \operatorname{Tan}^{-1} A-\operatorname{Tan}^{-1} B (c) \operatorname{Cot}^{-1} A+\operatorname{Cot}^{-1} B (d) \operatorname{Cot}^{-1} A-\operatorname{Cot}^{-1} B

Solve:8. \sec ^{4} 8-\tan ^{4} 8=3
Solve:8.  \sec ^{4} 8-\tan ^{4} 8=3
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Solve:8. \sec ^{4} 8-\tan ^{4} 8=3

Q.25 If y=\cos ^{-1} x then range is(a) [0 \pi] (b) [-11] (c) \left[-\frac{\pi}{2} \frac{\pi}{2}\right] (d) none of theseLahore Board 2006
Q.25 If  y=\cos ^{-1} x  then range is(a)  [0 \pi] (b)  [-11] (c)  \left[-\frac{\pi}{2} \frac{\pi}{2}\right] (d) none of theseLahore Board 2006
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Q.25 If y=\cos ^{-1} x then range is(a) [0 \pi] (b) [-11] (c) \left[-\frac{\pi}{2} \frac{\pi}{2}\right] (d) none of theseLahore Board 2006

7. Prove that\[\sin ^{-1} x+\sin ^{-1} y=\sin ^{-1}\left(x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right)\]
7. Prove that\[\sin ^{-1} x+\sin ^{-1} y=\sin ^{-1}\left(x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right)\]
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7. Prove that\[\sin ^{-1} x+\sin ^{-1} y=\sin ^{-1}\left(x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right)\]

Range of \operatorname{Cos} x is.A. [01] B. \left[\begin{array}{ll}-1 & 0\end{array}\right] C. [-11] D. \left[\begin{array}{ll}-1 & 2\end{array}\right]
Range of  \operatorname{Cos} x  is.A.  [01] B.  \left[\begin{array}{ll}-1 & 0\end{array}\right] C.  [-11] D.  \left[\begin{array}{ll}-1 & 2\end{array}\right]
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Range of \operatorname{Cos} x is.A. [01] B. \left[\begin{array}{ll}-1 & 0\end{array}\right] C. [-11] D. \left[\begin{array}{ll}-1 & 2\end{array}\right]

Q.7 \cos \left(\sec ^{-1} 1\right) is equal to :(a) 1(b) 0(c) 30^{\circ} (d) 2
Q.7  \cos \left(\sec ^{-1} 1\right)  is equal to :(a) 1(b) 0(c)  30^{\circ} (d) 2
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Q.7 \cos \left(\sec ^{-1} 1\right) is equal to :(a) 1(b) 0(c) 30^{\circ} (d) 2

Solve:1. \sin \theta+\cos 8=1 .
Solve:1.   \sin \theta+\cos 8=1 .
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Solve:1. \sin \theta+\cos 8=1 .

Find the principal values of:5. \sin \left(\arccos \frac{\sqrt{3}}{2}+\arcsin \frac{1}{2}\right)
Find the principal values of:5.  \sin \left(\arccos \frac{\sqrt{3}}{2}+\arcsin \frac{1}{2}\right)
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Find the principal values of:5. \sin \left(\arccos \frac{\sqrt{3}}{2}+\arcsin \frac{1}{2}\right)

If \cos \theta=\frac{1}{2} then \theta is:A. \frac{\pi}{6} B. \frac{-\pi}{6} C. \frac{5 \pi}{6} D. \frac{-5 \pi}{6}
If  \cos \theta=\frac{1}{2}  then  \theta  is:A.  \frac{\pi}{6} B.  \frac{-\pi}{6} C.  \frac{5 \pi}{6} D.  \frac{-5 \pi}{6}
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If \cos \theta=\frac{1}{2} then \theta is:A. \frac{\pi}{6} B. \frac{-\pi}{6} C. \frac{5 \pi}{6} D. \frac{-5 \pi}{6}

Q.15 The domain of y=\sin ^{-1} x is(a) -1 \leq x \leq 1 (b) -1<x<1 (c) -\frac{\pi}{2} \leq x \leq \frac{\pi}{2} (d) -\frac{\pi}{2}<x<\frac{\pi}{2} Lahore Board 2014; Gujranwala Board 2013; 2007
Q.15 The domain of  y=\sin ^{-1} x  is(a)  -1 \leq x \leq 1 (b)  -1<x<1  (c)  -\frac{\pi}{2} \leq x \leq \frac{\pi}{2} (d)  -\frac{\pi}{2}<x<\frac{\pi}{2} Lahore Board 2014; Gujranwala Board 2013; 2007
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Q.15 The domain of y=\sin ^{-1} x is(a) -1 \leq x \leq 1 (b) -1<x<1 (c) -\frac{\pi}{2} \leq x \leq \frac{\pi}{2} (d) -\frac{\pi}{2}<x<\frac{\pi}{2} Lahore Board 2014; Gujranwala Board 2013; 2007

Q.26 \sin \left(\sin ^{-1}\left(\frac{1}{2}\right)\right) equals(a) \frac{1}{2} (b) -\frac{1}{2} (c) \frac{\pi}{3} (d) \frac{\pi}{6} Lahore Board 2008
Q.26  \sin \left(\sin ^{-1}\left(\frac{1}{2}\right)\right)  equals(a)  \frac{1}{2} (b)  -\frac{1}{2} (c)  \frac{\pi}{3} (d)  \frac{\pi}{6} Lahore Board 2008
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Q.26 \sin \left(\sin ^{-1}\left(\frac{1}{2}\right)\right) equals(a) \frac{1}{2} (b) -\frac{1}{2} (c) \frac{\pi}{3} (d) \frac{\pi}{6} Lahore Board 2008

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