Question

verify the identity.
\\( \cos x+\cos x \tan ^{2} x=\sec x \\)
to verify the identity, start with the more complicated side and transform it to look like the other side. choose the com
\\( \cos x+\cos x \tan ^{2} x \\)
\\( =\cos x(\square) \\)
(do not simplify.)
\\( =\cos x(\square) \\)
\\( =\square \\)
(do not simplify.)
\\( =\sec x \\)

Explanation:

Step1: Factor out \(\cos x\)

$$ \cos x+\cos x\tan^{2}x=\cos x(1 + \tan^{2}x) $$

Step2: Use the Pythagorean identity \(1+\tan^{2}x=\sec^{2}x\)

$$ \cos x(1 + \tan^{2}x)=\cos x\sec^{2}x $$

Step3: Express \(\sec x\) as \(\frac{1}{\cos x}\)

$$ \cos x\sec^{2}x=\cos x\times\frac{1}{\cos^{2}x} $$

Step4: Simplify the expression

$$ \cos x\times\frac{1}{\cos^{2}x}=\frac{1}{\cos x}=\sec x $$

Answer:

The identity \(\cos x+\cos x\tan^{2}x = \sec x\) is verified.