Question

write the expression $\sec x(-\sin ^{2}x + 1)$ as a single term
a) $\cos x$
b) $-\sec ^{2}x$
c) $-\cos x$
d) $\cos ^{2}x$

Explanation:

Step1: Use Pythagorean identity

We know that \(\sin^{2}x+\cos^{2}x = 1\), so \(1-\sin^{2}x=\cos^{2}x\).
The given expression \(\sec x(-\sin^{2}x + 1)\) becomes \(\sec x\times\cos^{2}x\).

Step2: Use reciprocal identity

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

Step3: Simplify the expression

\(\frac{1}{\cos x}\times\cos^{2}x=\cos x\)

Answer:

A. \(\cos x\)