Prove the following identities: `(sin (2A + B))/sin A - 2 cos (A + B) = sin B/sin A`

Prove the following identities : (sin A + cos A)/ (sin A - cos A) + (sin A - cos A)/ (sin A + cos A) = (2)/ (2 sin^(2) A - 1)

Prove the following identities : (cos A + sin A)^(2) + (cos A - sin A)^(2) = 2

Prove the following identities : (sin(A+B) - 2 sin A + sin (A-B))/(cos(A+B)-2cos A +cos (A-B))=tan A

Prove each of the following identities : ((sin A - sin B))/((cos A + cos B)) + (( cos A - cos B))/((sin A + sin B)) =0

Prove the following identities : tan^(2) A - tan^(2) B = (sin^(2) A - sin^(2) B)/ (cos^(2) A . cos^(2) B)

Prove the following : (sin(A+B)+cos(A-B))/(sin(A-B)+cos(A+B)) = sec2B+tan2B

Prove the following identity : sin^2 A cos^2 B + cos^2 A sin^2 B + cos^2 A cos^2 B+ sin^2 A sin^2 B = 1 .

Prove the following : sin2A+sin2B+sin2(A-B) = 4sinA.cosB.cos(A-B)

Prove that : sin (A + B) sin (A - B) = sin^2A- sin^2B .

If A+B+C=pi/2 prove the following (i) sin 2A+sin 2B +sin 2C=4 cos A cos B cos C (ii) cos 2A +cos 2B+cos 2C=1+4 sin A sin B sin C.