Question

the expression below simplifies to a constant, a single function, or a power of a function. use fun

\\( \frac { \sin ^ { 2 } x } { \cos ^ { 2 } x } + \tan x \cot x \\)

\\( \frac { \sin ^ { 2 } x } { \cos ^ { 2 } x } + \tan x \cot x = \\)

Explanation:

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

We know that \(\frac{\sin x}{\cos x}=\tan x\), so \(\frac{\sin^{2}x}{\cos^{2}x}=\tan^{2}x\).

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

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

Step3: Combine the two simplified parts

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

Answer:

\(\sec^{2}x\)