Question

complete the sentence so the result is an identity. let x be any real number.
1 + ___ = csc²x
a. cot²x
b. sin²x
c. cos²x
d. sec²x

Explanation:

Step1: Recall the Pythagorean identity

We know the Pythagorean identity \(1+\cot^{2}x=\csc^{2}x\). This identity is derived from the basic trigonometric definitions. Since \(\csc x=\frac{1}{\sin x}\) and \(\cot x = \frac{\cos x}{\sin x}\), then \(\csc^{2}x-\cot^{2}x=\frac{1}{\sin^{2}x}-\frac{\cos^{2}x}{\sin^{2}x}=\frac{1 - \cos^{2}x}{\sin^{2}x}\). And using \(\sin^{2}x+\cos^{2}x = 1\) (i.e., \(1-\cos^{2}x=\sin^{2}x\)), we get \(\frac{\sin^{2}x}{\sin^{2}x}=1\), so \(1+\cot^{2}x=\csc^{2}x\).

Answer:

A. \(\cot^{2}x\)