How to simplify a trigonometric identity involving secant and tangent

Key concept: This kind of identity usually looks longer than it really is. The trick is spotting a difference of squares and then using the reciprocal relationship between secant and tangent.

Step by step

We want to prove

$\frac{\sin^4 x-\cos^4 x}{\sin^2 x-\cos^2 x}+\frac{1}{\sec x+\tan x}+\frac{1}{\sec x-\tan x}=2\cos^2 x.$

Step 1: Simplify the first fraction

Factor the numerator as a difference of squares:

$\sin^4 x-\cos^4 x=(\sin^2 x-\cos^2 x)(\sin^2 x+\cos^2 x).$

So

$\frac{\sin^4 x-\cos^4 x}{\sin^2 x-\cos^2 x}=\sin^2 x+\cos^2 x=1.$

Step 2: Simplify the secant-tangent terms

Use the identity

$\sec^2 x-\tan^2 x=1.$

Then

$\frac{1}{\sec x+\tan x}=\frac{\sec x-\tan x}{(\sec x+\tan x)(\sec x-\tan x)}=\sec x-\tan x,$

and similarly

$\frac{1}{\sec x-\tan x}=\sec x+\tan x.$

Add them:

$\left(\sec x-\tan x\right)+\left(\sec x+\tan x\right)=2\sec x.$

Step 3: Combine everything

The left-hand side becomes

$1+2\sec x.$

So the expression as written does not simplify to $2\cos^2 x$ in general.

That means there is likely a typo in the target identity or in one of the terms. If the intended expression was slightly different, the simplification could change.

What to check

• whether the numerator should be $\sin^4 x-\cos^4 x$,
• whether the right-hand side should be $1+2\sec x$,
• or whether one of the sec/tan terms has the opposite sign.

Pitfall alert

Do not force an identity to work just because it looks familiar. If you simplify carefully and get a different result, the original problem statement may contain an error.

Also, remember that $\frac{1}{\sec x\pm\tan x}$ is not arbitrary: it is usually simplified by multiplying by the conjugate, because $\sec^2 x-\tan^2 x=1$.

Try different conditions

If the expression were

$\frac{\sin^4 x-\cos^4 x}{\sin^2 x-\cos^2 x}+\frac{1}{\sec x+\tan x}-\frac{1}{\sec x-\tan x},$

then the sec-tan part would simplify to

$\left(\sec x-\tan x\right)-\left(\sec x+\tan x\right)=-2\tan x,$

and the whole expression would become $1-2\tan x$.

So the sign pattern matters a lot.

Further reading

difference of squares, secant and tangent, trigonometric simplification