Verifying trigonometric identities with Pythagorean formulas

Expert intro: Pythagorean identities such as $\sin^2\theta+\cos^2\theta=1$ drive nearly every line in this trigonometric proof set.

Detailed walkthrough

Core identities used throughout

The entire list is built from a small group of standard formulas: $\sin^2\theta+\cos^2\theta=1,$ $1+\tan^2\theta=\sec^2\theta,$ $1+\cot^2\phi=\csc^2\phi,$ and the algebraic expansion $(a+b)^2=a^2+2ab+b^2.$ Once those are available, each identity can be proved by rewriting one side until it matches the other.

For example, in part a, $(1-\sin\theta)(1+\sin\theta)=1-\sin^2\theta=\cos^2\theta,$ which is the difference of squares plus the Pythagorean identity.

In part b, $(1+\tan^2\theta)\cos^2\theta=\sec^2\theta\cos^2\theta=1,$ because $\sec\theta=1/\cos\theta$.

In part c, $(\sin A+\cos A)^2=\sin^2A+2\sin A\cos A+\cos^2A=1+2\sin A\cos A.$

How to handle the mixed forms

Expressions like $\cos^2x-\sin^2x$ and $1-2\sin^2x$ are often connected by the same core identity. For d, $\cos^2x-\sin^2x=(1-\sin^2x)-\sin^2x=1-2\sin^2x.$ That is just a substitution of $\cos^2x=1-\sin^2x$.

For e, $\tan^2\phi+\cot^2\phi = (\sec^2\phi-1)+(\csc^2\phi-1)=\sec^2\phi+\csc^2\phi-2,$ so the statement as written, $\tan^2\phi+\cot^2\phi=\csc^2\phi+1$, is not an identity. A quick test at $\phi=\pi/4$ gives $1+1=2$ on the left and $2+1=3$ on the right, so it fails.

For f, $3\cos^2\theta-2=3(1-\sin^2\theta)-2=1-3\sin^2\theta.$ And for g, $2\tan^2A-1=2(\sec^2A-1)-1=2\sec^2A-3.$

The reciprocal identities and the false item

Part h reads $1-\tan^2\theta+\sec^2\theta=2.$ Using $\sec^2\theta=1+\tan^2\theta$, the left side becomes $1-\tan^2\theta+1+\tan^2\theta=2,$ so h is true.

Part i, $2\sin^2x=1-2\cos^2x,$ looks unusual at first, but it is only a rearrangement of $\sin^2x=1-\cos^2x$: $1-2\cos^2x=2(1-\cos^2x)-1=2\sin^2x-1,$ so as written, i is also not generally true. A simple check at $x=0$ gives $0$ on the left and $-1$ on the right.

Part j, $\tan^2\theta = 1 + \tan^2\theta,$ is false because it would require $1=0$. It is most likely a typo in the source. The valid related identity is $1+\tan^2\theta=\sec^2\theta$.

What to remember when proving trig identities

The fastest proof strategy is to pick one side and rewrite it using a single core identity. If both sides are complicated, choose the side with the larger variety of functions and convert everything into sine and cosine, or into secant and tangent, but do not mix too many bases at once.

Also, not every printed “identity” is actually true. A quick numerical check at a convenient angle such as $\pi/4$ or $0$ can reveal a typo before you spend time proving something impossible. Here, parts e, i, and j are not valid as written, while a, b, c, d, f, g, and h are correct.

💡 Pitfall guide

The biggest trap is trying to prove every line the same way without checking whether each statement is actually true. In this set, some items are genuine identities, but others are not. A common mistake is to accept part e or part j just because it resembles a standard formula. Another issue is that you can accidentally rewrite the wrong side and introduce circular reasoning. For example, if you start from $\tan^2\theta=1+\tan^2\theta$, no algebraic manipulation will rescue it because the statement is false. Before proving, test suspicious expressions at a simple angle such as $0$ or $\pi/4$; that saves time and prevents you from building a proof on an incorrect premise.

🔄 Real-world variant

If part g were changed from $2\tan^2A-1=2\sec^2A-3$ to $2\tan^2A+1=2\sec^2A-1$, it would still be true, because substituting $\sec^2A=1+\tan^2A$ gives $2(1+\tan^2A)-1=2\tan^2A+1$. If part e were corrected to $\tan^2\phi+1=\sec^2\phi$, it would become the standard Pythagorean identity and could be proved immediately. A useful variant proof exercise is to verify $(\cos x-\sin x)^2=1-2\sin x\cos x$; this changes the sign in the middle term while keeping the same expansion pattern. These variants show how a small sign change can turn a false statement into a true identity or vice versa.

🔍 Related terms

Pythagorean identity, difference of squares, trigonometric reciprocity