Question

Prove that:sin3A/sinA - cos3A/cosA=2​

Answer 1

The result essentially follows directly from the triple-angle identities,

sin(3A) = 3 sin(A) cos²(A) - sin³(A)

cos(3A) = cos³(A) - 3 sin²(A) cos(A)

Then

sin(3A)/sin(A) = 3 cos²(A) - sin²(A)

cos(3A)/cos(A) = cos²(A) - 3 sin²(A)

and

sin(3A)/sin(A) - cos(3A)/cos(A)

= 3(cos²(A) + sin²(A)) - (sin²(A) + cos²(A))

= 3 - 1

= 2