How to verify (1 − cos x)(1 + cos x) = sin²x

Key concept: This identity is the cosine version of a very familiar pattern. Once you see the structure, the proof is almost automatic.

Step by step

Step 1: Use the difference of squares

Expand the product: $(1-\cos x)(1+\cos x)=1-\cos^2 x.$

Step 2: Replace using the Pythagorean identity

From $\sin^2 x+\cos^2 x=1,$ we get $1-\cos^2 x=\sin^2 x.$

Step 3: Conclude

So $(1-\cos x)(1+\cos x)=\sin^2 x,$ which verifies the identity.

Pitfall alert

A frequent slip is writing $1-\cos^2 x=\cos^2 x$. That is not correct. The Pythagorean identity says the leftover term is $\sin^2 x$. Keeping track of which function is being squared matters a lot here.

Try different conditions

If the expression were $(1+\cos x)^2-(\sin x)^2$, you could factor it as a difference of squares too: $(1+\cos x-\sin x)(1+\cos x+\sin x).$ That kind of rearrangement is useful when the goal is factoring rather than direct verification.

Further reading

difference of squares, Pythagorean identity, trig factoring