Question

Prove that (sinA* cosB + sinB * cosA)/(cosA * sinB + cosB * sinA) = 1

Answer 1

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Answer:

How do I prove that (cosA-cosB)/(sinA+sinB)=(sinB-sinA)/(cosA+cosB) using trigonometric identities formulas?

E=cos(A)−cos(B)sin(A)+sin(B)

Multiplying the numerator and denominator by cos(A)+cos(B)

E=cos(A)−cos(B)sin(A)+sin(B)×cos(A)+cos(B)cos(A)+cos(B)

=cos2(A)−cos2(B)[sin(A)+sin(B)][cos(A)+cos(B)]

Using the identities cos2(A)=1−sin2(A) and cos2(B)=1−sin2(B)

E=1−sin2(A)−1+sin2(B)[sin(A)+sin(B)][cos(A)+cos(B)]

=sin2(B)−sin2(A)[sin(A)+sin(B)][cos(A)+cos(B)]

=[sin(B)−sin(A)][sin(B)+sin(A)][sin(A)+sin(B)][cos(A)+cos(B)]

Dividing the numerator and denominator by sin(A)+sin(B) , we have:

E=sin(B)−sin(A)cos(A)+cos(B)