Question

if sin A = sin x + sin y / 1+ sin x - sin y ,then prove that cos A = cos x cos y / 1+ sin x sin y ​

Answer 1

Answer:

Given

sinθ=sinx+siny1+sinxsiny

show

cosθ=cosxcosy1+sinxsiny

This is surely not correct because the sine doesn’t determine the sign of the cosine, so there’s probably a missing ±; let’s find out.

We have cos2θ+sin2θ=1 or

cos2θ=1−sin2θ

and we just work it out. There’s a common denominator; it’s r2 where r=1+sinxsiny. We focus on the numerator:

r2cos2θ=r2−r2sin2θ

=(1+sinxsiny)2−(sinx+siny)2

=1+2sinxsiny+sin2xsin2y−sin2x−2sinxsiny−sin2y

=1+sin2xsin2y−sin2x−sin2y

=1+(1−cos2x)(1−cos2y)−(1−cos2x)−(1−cos2y)

=1+1−cos2x−cos2y+cos2xcos2y−1+cos2x−1+cos2y

=cos2xcos2y

cos2θ=cos2xcos2y(1+sinxsiny)2

cosθ=±cosxcosy1+sinxsiny

Step-by-step explanation:

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