Question

simplify.
\sin^{2}x \sec^{2}x - \sec^{2}x

Explanation:

Step1: Factor out \(\sec^{2}x\)

We can factor \(\sec^{2}x\) from both terms in the expression \(\sin^{2}x\sec^{2}x - \sec^{2}x\). Using the distributive property \(ab - ac=a(b - c)\), where \(a = \sec^{2}x\), \(b=\sin^{2}x\) and \(c = 1\), we get:
\(\sec^{2}x(\sin^{2}x - 1)\)

Step2: Use the Pythagorean identity

Recall the Pythagorean identity \(\sin^{2}\theta+\cos^{2}\theta = 1\), which can be rearranged as \(\sin^{2}\theta- 1=-\cos^{2}\theta\). Substituting \(\theta=x\) into our expression, we have \(\sin^{2}x - 1=-\cos^{2}x\). So our expression becomes:
\(\sec^{2}x(-\cos^{2}x)\)

Step3: Recall the definition of \(\sec x\)

The secant function is defined as \(\sec x=\frac{1}{\cos x}\), so \(\sec^{2}x=\frac{1}{\cos^{2}x}\). Substituting this into our expression:
\(\frac{1}{\cos^{2}x}\times(-\cos^{2}x)\)

Step4: Simplify the product

Multiplying \(\frac{1}{\cos^{2}x}\) and \(-\cos^{2}x\), the \(\cos^{2}x\) terms cancel out, leaving us with \(- 1\).

Answer:

\(-1\)