RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2

Rajasthan Board RBSE Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2 Textbook Exercise Questions and Answers.

RBSE Class 12 Maths Solutions Chapter 2 Inverse Trigonometric Functions Ex 2.2

Question 1.
3 sin^-1 x = sin^-1 (3x - 4x³), x ∈ \(\left[-\frac{1}{2}, \frac{1}{2}\right]\)
Answer:
Let sin^-1 x = θ ⇒ sin θ = x
∵ sin 3θ = 3 sin θ - 4 sin³θ
⇒ sin 3θ = 3x - 4x³
⇒ 3θ = sin^-1(3x - 4x³)
Thus, 3 sin^-1 x = sin^-1 (3x - 4x³)
Hence Proved.

Question 2.
3 cos^-1x = cos^-1 (4x³ - 3x), x ∈ \(\left[\frac{1}{2}, 1\right]\)
Answer:
Let cos^-1 x = θ ⇒ cos θ = x
⇒ cos 3θ = 4 cos 3θ - 3 cos θ
⇒ cos 3θ = 4x³ - 3x
⇒ 3θ = cos^-1 (4x³ - 3x)
Thus, 3 cos^-1 x = cos^-1 (4x³ - 3x)
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 functions in the simplest form:

Question 5.
tan^-1\(\frac{\sqrt{1+x^{2}}-1}{x}\), x ≠ 0
Answer:

Question 6.
tan^-1\(\frac{1}{\sqrt{x^{2}-1}}\), |x| > 1
Answer:

Question 7.
tan^-1\(\left(\sqrt{\frac{1-\cos x}{1+\cos x}}\right)\), 0 < x < π
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:

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, - \(\frac{a}{\sqrt{3}}\) < x < \(\frac{a}{\sqrt{3}}\)
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:
We have, cot (tan^-1 a + cot^-1 a)
= cot \(\frac{\pi}{2}\) = 0 (∵ tan^-1 a + cot^-1 a = \(\frac{\pi}{2}\))
Hence, cot(tan^-1 a + cot^-1 a) = 0

Question 13.
\(\tan \frac{1}{2}\left[\sin ^{-1} \frac{2 x}{1+x^{2}}+\cos ^{-1} \frac{1-y^{2}}{1+y^{2}}\right]\), |x| < 1, y > 0 and xy > 1
Answer:

Question 14.
If sin(sin^-1 \(\left(\frac{1}{5}\right)\) + cos^-1 x) = 1, then find the value of x.
Answer:

Question 15.
If tan^-1\(\left(\frac{x-1}{x-2}\right) \)+ tan^-1\(\left(\frac{x+1}{x+2}\right)\) = \(\frac{\pi}{4}\), then find value of x.
Answer:

Find the values of each of the expression in questions 16 to 18:

Question 16.
sin^-1 (sin\(\frac{2 \pi}{3}\)).
Answer:

Question 17.
tan^-1 \(\left(\tan \frac{3 \pi}{4}\right)\)
Answer:

Question 18.
tan \(\left(\sin ^{-1} \frac{3}{5}+\cot ^{-1} \frac{3}{2}\right)\)
Answer:

Question 19.
cos^-1 \(\left(\cos \frac{7 \pi}{6}\right)\) is equal to:
(A) \(\frac{7 \pi}{6}\)
(B) \(\frac{5 \pi}{6}\)
(C) \(\frac{\pi}{3}\)
(D) \(\frac{\pi}{6}\)
Answer:
The principal value branch of cos^-1 is [0, π].

Thus, option (B) is correct.

Question 20.
sin [\(\frac{\pi}{3}\)-sin^-1 \(\left(-\frac{1}{2}\right)\)] is equal to:
(A) \(\frac{1}{2}\)
(B) \(\frac{1}{3}\)
(C) \(\frac{1}{4}\)
(D) 1
Answer:

Thus, option(D) is correct.

Question 21.
tan^-1 √3 - cot^-1 (- √3) is equal to:
(A) π
(B) - \(\frac{\pi}{2}\)
(C) 0
(D) 2√3
Answer: