Question

1+sinx+cosx/1+sinx-cosx=cotx/2 ​

Answer 1

Answer:

LHS = (1-sin x + cos x)/(1-sin x - cos x) or

= [(cos^2 (x/2) + sin^2 (x/2) - 2 sin (x/2)cos (x/2) + cos^2 (x/2) - sin^2 (x/2)] / [(cos^2 (x/2) + sin^2 (x/2) - 2 sin (x/2)cos (x/2) - cos^2 (x/2) + sin^2 (x/2)], or

= [2 cos^2 (x/2) - 2 sin (x/2)cos (x/2)]/[2 sin^2 (x/2) - 2 sin (x/2)cos (x/2)], or

= 2cos (x/2)[cos (x/2) - sin(x/2)] / 2[sin (x/2)[-cos (x/2) + sin(x/2)], or

= -2cos (x/2)[cos (x/2) - sin(x/2)] / 2[sin (x/2)[cos (x/2) - sin(x/2)], or

= - cos (x/2)/ sin (x/2)

= - cot (x/2). Proved.

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