# Why is $2\cos^210^\circ -2\cos^220^\circ =\sin10^\circ$?

I was to simplify an expression, I came to this:

$$X=2\cos^210^\circ -2\cos^220^\circ$$

But, I couldn't simplify it further.

I checked the answer and it was $\sin10^\circ$. But I can't understand why.

Can anyone help?

EDIT: $\theta$ was $10^\circ$. I myself generalized it.

The two sides are not equivalent. So $10^\circ$ seems to have a special characteristic which we should use here.

• Check $\theta =\frac{\pi}{4}$. – Rohan Dec 30 '16 at 11:18
• There's a mistake somewhere ; LHS is even, RHS is odd. What was the expression to simplify? – D. Thomine Dec 30 '16 at 11:19
• @Fib1123 It was 10. – AHB Dec 30 '16 at 11:23
• @D.Thomine Yes. It was an equation. Not an Identity. Edited. – AHB Dec 30 '16 at 11:33

Use the “sum to product formulas” \begin{align} \cos10^\circ+\cos20^\circ&=2\cos15^\circ\cos5^\circ\\ \cos10^\circ-\cos20^\circ&=2\sin15^\circ\sin5^\circ \end{align} so your expression is $$8\sin15^\circ\cos15^\circ\sin5^\circ\cos5^\circ= 2\sin30^\circ\sin10^\circ=\sin10^\circ$$ In general, $$2(\cos^2\theta-\cos^22\theta)= 8\sin\frac{3\theta}{2}\cos\frac{3\theta}{2} \sin\frac{\theta}{2}\cos\frac{\theta}{2} = 2\sin3\theta\sin\theta$$