Prove that (sin7x + sin5x) + (sin9x + sin3x)/(cos7x + cos5x) + (cos9x + cos3x) = tan6x

Question

Prove that

\(\cfrac{(sin7x\,+\,sin5x) +(sin9x\,+\,sin3x)}{(cos7x\,+\,cos5x)\,+\,(cos9x\,+\,cos3x)}=tan6x\)

(sin7x + sin5x) + (sin9x + sin3x)/(cos7x + cos5x) + (cos9x + cos3x) = tan6x

Answer 1

\(\cfrac{(sin7x\,+\,sin5x) +(sin9x\,+\,sin3x)}{(cos7x\,+\,cos5x)\,+\,(cos9x\,+\,cos3x)}\)

= \(\cfrac{sin6x}{cos6x}\)

= tan6x

Using the formula,

sinA + sinB = 2sin\(\frac{A\,+\,B}{2}\)cos\(\frac{A\,-\,B}{2}\)

cosA + cosB = 2cos\(\frac{A\,+\,B}{2}\)cos\(\frac{A\,-\,B}{2}\)