simplify: (cos x)^2(tan x)^2 - 1 / (cos x)^2

Answer:

$\tan^{2}x - \sec^{2}x$

Explanation:

Step1: Split the fraction

$\frac{(\cos x)^{2}(\tan x)^{2}}{(\cos x)^{2}}-\frac{1}{(\cos x)^{2}}$

Step2: Simplify the first - term

$\frac{(\cos x)^{2}(\tan x)^{2}}{(\cos x)^{2}}=\tan^{2}x$

Step3: Recall the secant definition

Since $\sec x=\frac{1}{\cos x}$, then $\frac{1}{(\cos x)^{2}}=\sec^{2}x$

Step4: Get the result

The simplified form is $\tan^{2}x - \sec^{2}x$