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Question

sinx-sinxxvl +sinx -V1- sinx4

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Solution

We have to prove that cot 1 ( 1+sinx + 1sinx 1+sinx 1sinx )= x 2 ,x( 0, π 4 ).

Solve left hand side of the given equation by rationalizing the angle.

cot 1 ( 1+sinx + 1sinx 1+sinx 1sinx )= cot 1 ( ( 1+sinx + 1sinx ) 2 ( 1+sinx ) 2 ( 1sinx ) 2 ) = cot 1 ( 1+sinx+1sinx+2 ( 1+sinx )( 1sinx ) 1+sinx1+sinx ) = cot 1 ( 2( 1+ 1 sin 2 x ) 2sinx ) = cot 1 ( 1+cosx sinx )

Further solving,

cot 1 ( 1+sinx + 1sinx 1+sinx 1sinx )= cot 1 ( 2 cos 2 x 2 2sin x 2 cos x 2 ) = cot 1 ( cot x 2 ) = x 2

Hence, it is proved that cot 1 ( 1+sinx + 1sinx 1+sinx 1sinx )= x 2 .


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