How to rewrite cos²θ − sin²θ in another trig form

Key takeaway: This expression is one of those that can be read in more than one way. Depending on what you want, you may leave it as is, or rewrite it into a form that shows a different trig identity.

Key observation

Start with $\cos^2\theta-\sin^2\theta.$

A useful rewrite comes from the Pythagorean identity: $\cos^2\theta=1-\sin^2\theta.$ Substitute that into the left side: $\cos^2\theta-\sin^2\theta=(1-\sin^2\theta)-\sin^2\theta.$

Simplify

$1-\sin^2\theta-\sin^2\theta=1-2\sin^2\theta.$

So $\cos^2\theta-\sin^2\theta=1-2\sin^2\theta.$

You can also write the same expression as $2\cos^2\theta-1$ by using $\sin^2\theta=1-\cos^2\theta$ instead.

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Pitfalls the pros know 👇 A lot of errors here come from treating $\cos^2\theta-\sin^2\theta$ as if it were $\cos\theta-\sin\theta$. The squares belong to the individual trig values. Another common slip is forgetting that this identity is just one of several equivalent forms of $\cos 2\theta$.

What if the problem changes? If you replace the left side with $\sin^2\theta-\cos^2\theta$, the result becomes $-(\cos^2\theta-\sin^2\theta)$, which is $2\sin^2\theta-1$. In other words, the sign flips the whole expression.

Tags: double-angle identity, Pythagorean identity, equivalent forms