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## 2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2

### 2nd PUC Maths Inverse Trigonometric Functions NCERT Text Book Questions and Answers Ex 2.2

Question 1.

\(3 \sin ^{-1} x=\sin ^{-1}\left(3 x-4 x^{3}\right), \quad x \in\left[-\frac{1}{2}, \frac{1}{2}\right]\)

Answer:

Let sin^{-1} x be θ ⇒ x = sin θ

In RHS putting the value of x

sin^{-1} (3x – 4x^{3}) ⇒ sin^{-1} (3 sinθ – 4 sin^{3}θ)

= sin^{-1} (sin 3θ)

[∵ 3sinθ – 4sin^{3} θ = sin3θ]

= 3θ

putting the value of θ

3θ = 3 sin^{-1} x = LHS

∴ 3sin^{-1} x = sin^{-1} (3x – 4x^{3})

Hence proved.

Question 2.

\(3 \cos ^{-1} x=\cos ^{-1}\left(4 x^{3}-3 x\right), x \in\left[\frac{1}{2}, 1\right]\)

Answer:

Let cos^{-1} x be θ ⇒ x = cos θ

In RHS, putting the value of x

cos^{-1} (4x^{3} – 3x) ⇒ cos^{1} (4cos^{3} θ – 3 cosθ)

⇒ cos^{-1} (cos 3θ)

[∵ 4cos^{3} θ – 3cosθ = cos 3θ]

= 30

putting the value of 0 = 30

⇒ 3 cos^{-1} x = LHS

LHS = RHS

Hence proved

Question 3.

\(\tan ^{-1} \frac{2}{11}+\tan ^{-1} \frac{7}{24}=\tan ^{-1} \frac{1}{2}\)

Answer:

Question 4.

\(2 \tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{7}=\tan ^{-1} \frac{31}{17}\)

Answer:

Write the following function in the simplest form:

Question 5.

\(\tan ^{-1} \frac{\sqrt{1+x^{2}}-1}{x}, x \neq 0\)

Answer:

Question 6.

\(\tan ^{-1} \frac{1}{\sqrt{x^{2}-1}},|x|>1\)

Answer:

Let cosec^{-1} x = 0

⇒ x = cosec 0

Question 7.

\(\tan ^{-1}(\sqrt{\frac{1-\cos x}{1+\cos x}}), 0<x<\pi\)

Answer:

Question 8.

\(\tan ^{-1}\left(\frac{\cos x-\sin x}{\cos x+\sin x}\right), \frac{-\pi}{4}<x<\frac{3 \pi}{4}\)

Answer:

Divide each term of numerator and denominator inside the brackets by cos x

\(\Rightarrow \tan ^{-1}\left[\frac{(\cos x-\sin x) / \cos x}{(\cos x+\sin x) / \cos x}\right] \Rightarrow \tan ^{-1}\left[\frac{1-\tan x}{1+\tan x}\right]\)

Question 9.

\(\tan ^{ -1 } \frac { { x } }{ \sqrt { { a }^{ 2 }-{ x }^{ 2 } } } ,|{ x }|<{ a }\)

Answer:

Question 10.

\(\tan ^{-1}\left(\frac{3 a^{2} x-x^{3}}{a^{3}-3 a x^{2}}\right), a>0 ; \frac{-a}{\sqrt{3}} \leq x \leq \frac{a}{\sqrt{3}}\)

Answer:

Find the values of each of the following

Question 11.

\(\tan ^{-1}\left[2 \cos \left(2 \sin ^{-1} \frac{1}{2}\right)\right]\)

Answer:

Question 12.

cot (tan^{-1} a + cot^{-1} a)

Answer:

Question 13.

\(\begin{aligned}&\tan \frac{1}{2}\left[\sin ^{-1} \frac{2 x}{1+x^{2}}+\cos ^{-1} \frac{1-y^{2}}{1+y^{2}}\right]\\&|x|<1, y>0 \text { and } x y<1\end{aligned}\)

Answer:

Question 14.

In \(\sin \left(\sin ^{-1} \frac{1}{5}+\cos ^{-1} x\right)=1\)

Answer:

Question 15.

\(\text { If } \tan ^{-1} \frac{x-1}{x-2}+\tan ^{-1} \frac{x+1}{x+2}=\frac{\pi}{4}\)Then find the value of x

Answer:

Find the values of each of the expressions in Exercises 16 to 18.

Question 16.

\(\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)\)

Answer:

Question 17.

\( \tan ^{-1}\left(\tan \frac{3 \pi}{4}\right)\)

Answer:

Question 18.

\(\tan \left(\sin ^{-1} \frac{3}{5}+\cot ^{-1}\left(\frac{3}{2}\right)\right)\)

Answer:

Choose the correct Answer:

Question 19.

\(\cos ^{-1}\left(\cos \frac{7 \pi}{6}\right) \text { is equal to }\)

(A) \(\frac{7 \pi}{6}\)

(B) \(\frac{5 \pi}{6}\)

(C) \(\frac{\pi}{3}\)

(D) \(\frac{\pi}{6}\)

Answer:

Question 20.

\(\sin \left(\frac{\pi}{3}-\sin ^{-1}\left(-\frac{1}{2}\right)\right)\)is equal to

(A) π

(B) \(-\frac{\pi}{2}\)

(C) 0

(D) \(2 \sqrt{3}\)

Answer:

Question 21.

\(\tan ^{-1} \sqrt{3}-\cot ^{-1}(-\sqrt{3})\) is equal to

(A) π

(B) \(-\frac{\pi}{2}\)

(C) 0

(D) \(2 \sqrt{3}\)

Answer: