Choose the correct answer

is equal to

We need to find the value of

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

Using trigonometric identity,

[∵, lies in 1^{st} Quadrant and sine, cosine, tangent and cot are positive in 1^{st} Quadrant]

Using property of inverse trigonometry,

cot^{-1}(cot x) = x

Using trigonometric identity,

Substituting this value of ,

We had let above that cos^{-1} x = y.

If,

cos^{-1} x = y

⇒ x = cos y

Therefore,

1