sin[cot^-1{tan(cos^-1 x)}] is equal to A. x B. √(1 - x^2) C. 1/x

Question

sin[cot^-1{tan(cos^-1x)}] is equal to

A. x 

B. \(\sqrt{1-x^2}\)

C. \(\frac{1}{\text x}\)

D. none of these

Answer 1

Correct option is A. x

We need to find the value of

sin [cot^-1 {tan (cos^-1x)}] …(i)

We can solve such equation by letting the inner most trigonometric function (here, cos^-1 x) as some variable, and solve systematically following BODMAS rule and other trigonometric identities.

Let cos^-1 x = y

We can re-write the equation (i),

sin [cot^-1 {tan (cos^-1x)}] = sin [cot^-1 {tan y}]

Using trigonometric identity,

tan y = cot\((\frac{\pi}2-y)\)

[\(\because\),  cot\((\frac{\pi}2-y)\) lies in 1^st Quadrant and sine, cosine, tangent and cot are positive in 1^st Quadrant]