Answer :

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 1st Quadrant and sine, cosine, tangent and cot are positive in 1st 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,


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