Question

the expression below simplifies to a constant, a single function, or a power of a function. use fundamental identities to simplify the expression.
\frac{\sin ^{2} x}{\cos ^{2} x}+\cot x \tan x
\frac{\sin ^{2} x}{\cos ^{2} x}+\cot x \tan x=\square

Explanation:

Step1: Simplify \(\frac{\sin^{2}x}{\cos^{2}x}\)

Using the identity \(\tan x=\frac{\sin x}{\cos x}\), so \(\frac{\sin^{2}x}{\cos^{2}x}=\tan^{2}x\).

Step2: Simplify \(\cot x\tan x\)

Since \(\cot x = \frac{1}{\tan x}\), then \(\cot x\tan x=\frac{1}{\tan x}\times\tan x = 1\).

Step3: Combine the two simplified parts

The original expression \(\frac{\sin^{2}x}{\cos^{2}x}+\cot x\tan x\) becomes \(\tan^{2}x + 1\).
Using the Pythagorean identity \(\tan^{2}x+1=\sec^{2}x\).

Answer:

\(\sec^{2}x\)