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,