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$\pi$].
$\endgroup$2 Answers
$\begingroup$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.
$\endgroup$ 1 $\begingroup$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 $(\cos\theta x-1)(\cos\theta x+1)=0$
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