Evaluate. tan 1/2x =csc x-cot x

PoentWeptgj 2022-08-01 Answered
Evaluate.
tan 1 / 2 x = csc x cot x
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Answers (1)

Quenchingof
Answered 2022-08-02 Author has 14 answers
tan 1 / 2 x = csc x cot x
sin ( A + B ) = sin A cos B + cos A sin B   a n d   cos ( A + B ) = cos A cos B sin A sin B
let B = A then sin ( 2 A ) = 2 sin A cos A ..... (1) and
cos ( 2 A ) = cos 2 A sin 2 A = ( 1 sin 2 A ) sin 2 A = 1 2 sin 2 A 2 sin 2 A = 1 cos ( 2 A ),,,,,,(2)
tan ( x / 2 ) = [ sin ( x / 2 ) ] / [ cos ( x / 2 ) ] = [ 2 sin ( x / 2 ) sin ( x / 2 ) ] / [ 2 sin ( x / 2 ) cos ( x / 2 ) ]
multiply numerator and denominator by 2 sin ( x / 2 )
= [ 1 cos x ] / [ sin x ] use (1) and (2)
= [ 1 / sin x ] [ cos x / sin x ] = csc x cot x
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