Maths - Please Help?

Question

Hello,

There are some questions that I am not sure how to tackle:

1) sin(A + B) sin(A - B) = sin^2A - sin^2B

2) (cos2A) / (cosA - sinA) = cosA + sinA

3) (sinA) / (sinB) + (cosA) / (cosB) = [2sin(A + B)] / [sin2B]

Thank you in advance.

Answer 1

1).

The trigonometric equation is sin(A + B)sin(A - B) = sin² A - sin² B.

Left hand side identity : sin(A + B)sin(A - B).

Composite formula : sin(A + B) = sin A cos B + cos A sin B,

                                   sin(A - B) = sin A cos B - cos A sin B.

= [ sin A cos B + cos A sin B ] * [ sin A cos B - cos A sin B ]

= sin² A cos² B - sin² B cos² A

Pythagorean identity : sin² A + cos² A = 1.

= sin² A (1 - sin² B)  - sin² B (1 - sin² A)

= sin² A - sin² Asin² B  - sin² B + sin² Bsin² A

= sin² A - sin² B

= Right hand side identity.

Hence proved.

2).

The trigonometric equation is cos(2A) / (cos A - sin A)= cos A + sin A.

Left hand side identity : cos(2A) / (cos A - sin A).

Double angle formula : cos(2A) = cos² A - sin² A.

= [ cos² A - sin² A ] / (cos A - sin A)

= [ (cos A + sin A)(cos A - sin A) ] / (cos A - sin A)

= cos A + sin A

= Right hand side identity.

Hence proved.

3).

The trigonometric equation is (sin A/sin B) + (cos A/cos B)= 2sin (A + B) / sin 2B.

Left hand side identity : (sin A/sin B) + (cos A/cos B).

= (sin A cos B + cos A sin B) / sin B cos B

Composite formula : sin(A + B) = sin A cos B + cos A sin B.

= sin(A + B) / sin B cos B

Multiply numerator and denominator by 2.

= 2sin(A + B) / 2sin B cos B

Double angle formula : sin(2B) = 2sin B cos B.

= 2sin(A + B) / sin (2B)

= Right hand side identity.

Hence proved.