How to simplify a trigonometric identity with secant and cosecant

Expert intro: This is a classic identity-check problem. The main move is to rewrite everything in terms of sine and cosine, then compare both sides carefully.

Detailed walkthrough

Start with the left-hand side:

$\frac{\cos x}{\sin x}+\frac{\sin x}{\cos x}$

Find a common denominator:

$=\frac{\cos^2 x+\sin^2 x}{\sin x\cos x}$

Use the Pythagorean identity:

$\cos^2 x+\sin^2 x=1$

So the expression becomes

$\frac{1}{\sin x\cos x}$

Now rewrite in reciprocal form:

$\frac{1}{\sin x\cos x}=\csc x\sec x$

So the identity is true wherever both sides are defined.

💡 Pitfall guide

A common mistake is canceling across a sum, like trying to simplify $\frac{\cos x}{\sin x}+\frac{\sin x}{\cos x}$ by combining numerators directly. Also, remember the expression is undefined when $\sin x=0$ or $\cos x=0$, so the identity is only valid on its domain.

🔄 Real-world variant

If the problem were written as $\tan x+\cot x$, the same idea still works:

$\tan x+\cot x=\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}=\frac{1}{\sin x\cos x}=\csc x\sec x$

The algebra changes a little, but the reciprocal form at the end is the same.

🔍 Related terms

trig identities, cosecant, secant