Power Reducing Formula

Question

I need some how with the power reducing formula. I'm having trouble understanding how to apply it. Here's an equation that utilizes it:

$\cos^2 (\theta x) - 1 = 0$

How do I solve this on the interval [0, 2$\pi$].

Answer 1

The power reduction formula for cosine is $\cos^2n=\frac12+\frac{\cos2n}2$. If we let $n=\theta x$, we get

$$\frac12\cos2\theta x-\frac12=0$$ $$\cos2\theta x=1$$ $$2\theta x=2k\pi$$

Although this equation probably doesn't need a power reduction formula as it can also be rewritten as $-\sin^2\theta x=0$ or $(\cos\theta x-1)(\cos\theta x+1)=0$

Answer 2

Use the basic trigonometric formula

$$\cos 2x=\cos^2x-\sin^2x=2\cos^2x-1$$

to get

$$\cos^2(x\theta)-1=\frac{\cos (2x\theta)-1}{2}$$

Thus

$$\cos^2(x\theta)-1=0\iff \cos(2x\theta)=1\iff 2x\theta=2k\pi\,,\,k\in\Bbb Z\ldots$$

Now end the exercise.