Proving Trigonometric Identities

For some problems involving trigonometric functions, it can be easier to solve them if we can change a trigonometric expression into an equivalent one with slightly different properties.

For example, if we had the integral

\int \cos{\left(3x\right)}\sin{\left(x\right)}dx

we could apply integration by substitution to solve this if we could convert $\cos\left(3x\right)$ into an expression that contained only $\cos{\left(x\right)}$ , even if they were raised to a higher power.

There is in fact an identity that we could use to do this

\cos{\left(3x\right)} = 4\cos^3{\left(x\right)} − 3\cos{\left(x\right)}

but we should be able to prove that this is indeed true before we use it.

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