Problem 80 - Algebra and Trigonometry

Question

Verify the identity: $$ \csc x \cos ^{2} x+\sin x=\csc x $$

Step-by-Step Solution

Answer

The given identity \( \csc x \cos^{2} x + \sin x = \csc x \) is verified.

Step 1: Write the Trigonometric Functions in Terms of Sine and Cosine

The first step involves writing the trigonometric functions in terms of sine and cosine. Here, the cosecant function can be expressed as the reciprocal of sine. So, rewrite the equation as \( \frac{1}{\sin x} \cos^{2} x + \sin x = \frac{1}{\sin x} \).

Step 2: Simplify Expression

Simplify the expression as \( \frac{\cos^{2} x}{\sin x} + \sin x = \frac{1}{\sin x} \). Then carry out the addition on the left side to get \( \frac{\cos^{2} x + \sin^{2} x}{\sin x} = \frac{1}{\sin x} \).

Step 3: Utilize the Pythagorean Identity

Apply the Pythagorean identity \( \cos^{2} x + \sin^{2} x = 1 \), so the equation simplifies to \( \frac{1}{\sin x} = \frac{1}{\sin x} \). This verifies the given identity.