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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 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,