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Question

PROVE IT

cosA-sinA+1/cosA + sinA-1 = cosecA = cotA, using identity

cosec*2A = 1 + cot*2A

Answer 1

we know cos A = 2 cos^2 A/2 - 1

so cos A + 1 = 2 cos^2 A/2

again cos A = 1 - 2 sin ^2 A/2

so cos A -  = 2 sin ^ 2 A/2

and sin A = 2 sin A/2 cos A/2

so

(cosA-sinA+1) = 2 cos^2 A/2 + 2 cos A/2 sin A/2 = 2 cos A/2(cos A/2- sin A/2)

cosA+sinA-1) = 2 cos A/2 sin A/2 - 2 sin ^2 A/2 = 2 sin A/2(cos A/2- sin A/2)

so (cosA-sinA+1)/(cosA+sinA-1) = (cos A/2)/ sin A/2

= (2 cos A/2 sin A/2)/( 2 sin ^2 A/2)

= sin A/(1- cos A)

= sin A(1+cosA)/(1- cos^2 A)

= sin A(1+cos A)/sin ^2 A

= (1+cosA)/sin A

= cosec A + sin A

identity cosec*2A = 1 + cot*2A is not used