Prove that $\cos^4(\theta)-\sin^4(\theta)=\cos(2\theta)$

The problem is to prove that $$\cos^4(\theta)-\sin^4(\theta)=cos(2\theta)$$.

So, here are my steps so far:

$$(\cos^2\theta)^2-(\sin^2\theta)^2=\cos(2\theta)$$

$$(\cos^2\theta+\sin^2\theta)(\cos^2\theta-\sin^2\theta)=\cos(2\theta)$$

$$(\cos^2\theta+\sin^2\theta)(\cos\theta+\sin\theta)(\cos\theta-\sin\theta)=\cos(2\theta)$$

I don't know where to go from here. Please help! I feel like I'm missing something super obvious but I don't know what...

Well, $$\cos^2(\theta)+\sin^2(\theta)=1$$, and $$\cos(2\theta)=\cos^2(\theta)-\sin^2(\theta)$$ is a standard Double-Angle identity. Can you continue from there?

• Oh ya! I can't believe I forgot that....I got the answer.
– user680251
Jun 9 '19 at 23:35

There are a couple of nice ways we can do it. Firstly, note that: $$\cos^4(\theta)-\sin^4(\theta)=\bigl(\cos^2\theta-\sin^2\theta\bigr)\bigl(\cos^2\theta+\sin^2\theta\bigr)$$ and since: $$\cos^2\theta+\sin^2\theta=1\tag{1}$$ we can say that: $$\cos^4\theta-\sin^4\theta=\cos^2\theta-\sin^2\theta=\cos2\theta$$ which is the double angle formula for cosine. This can be proved geometrically a number of ways but also as shown below

We can also take advantage of De Moivre's theorem, which states that: $$\left(\cos x+i\sin x\right)^n=\cos(nx)+i\sin(nx)\tag{2}$$ Using the fact that: $$e^{ix}=\cos(x)+i\sin(x)\tag{3}$$ We can do this by saying: $$\cos(2x)+i\sin(2x)=\left(\cos x+i\sin x\right)^2$$ $$\cos(2x)+i\sin(2x)=\cos^2x+2i\cos x\sin x+i^2\sin^2x$$ $$\cos(2x)+i\sin(2x)=\cos^2x-\sin^2x+2i\cos x\sin x$$ Now if we separate the real and imaginary parts we are left with: $$\cos(2x)=\cos^2x-\sin^2x,\,\sin(2x)=2\sin(x)\cos(x)\tag{4}$$

We have $$\cos^4(\theta)-\sin^4(\theta) = (\cos^2\theta+\sin^2\theta)(\cos^2\theta-\sin^2\theta)$$ where $$(\cos^2\theta+\sin^2\theta) = 1$$ and $$(\cos^2\theta-\sin^2\theta) = 2\cos^2(\theta)-1 = \cos(2\theta)$$

Hint:$$\cos(t) = \frac{e^{it} + e^{-it}}{2}$$ and $$\sin(t) = \frac{e^{it} - e^{-it}}{2i}$$

• I just remembered the Double-Angle Identity.
– user680251
Jun 10 '19 at 3:28