Question

complete the sentence so the result is an identity. let x be any real number. __ + tan²x = sec²x
a. cos²x
b. sin²x
c. - 1
d. 1

Explanation:

Step1: Recall trigonometric identity

We know the fundamental Pythagorean - trigonometric identity $\sin^{2}x+\cos^{2}x = 1$. Divide both sides of this identity by $\cos^{2}x$: $\frac{\sin^{2}x}{\cos^{2}x}+\frac{\cos^{2}x}{\cos^{2}x}=\frac{1}{\cos^{2}x}$.

Step2: Simplify the fractions

Since $\frac{\sin x}{\cos x}=\tan x$ and $\frac{1}{\cos x}=\sec x$, we have $\tan^{2}x + 1=\sec^{2}x$.

Answer:

D. 1