Multiply; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent.

Question

Quizplus · Accepted Answer

When you multiply, you get:

$$\frac{(\sec	heta - 	an	heta)(\sec	heta + 	an	heta)}{\sec^2	heta - 	an^2	heta} = \frac{1}{\cos	heta}$$

Using the fundamental identities $\sec^2	heta = 1+	an^2	heta$ and $\cos^2	heta = 1 - \sin^2	heta$, we can simplify both the numerator and denominator:

$$\frac{\left(\frac{1}{\cos	heta} - \frac{\sin	heta}{\cos	heta}ight)\left(\frac{1}{\cos	heta} + \frac{\sin	heta}{\cos	heta}ight)}{\frac{1}{\cos^2	heta} - \frac{\sin^2	heta}{\cos^2	heta}} =\frac{1}{\cos	heta} $$

Simplifying the numerator further:

$$\frac{\frac{1}{\cos^2	heta} - \frac{\sin^2	heta}{\cos^2	heta}}{\frac{1-\sin^2	heta}{\cos^2	heta}} =\frac{1}{\cos	heta} $$

$$\frac{1-\sin^2	heta}{\cos^2	heta}=\frac{\cos^2	heta}{\cos^2	heta} = 1 $$

$$\cos^2	heta + \sin^2	heta = 1 $$

The answer that is not equivalent is D, which states $	heta = \frac{\pi}{6}$. Plugging this value into the original equation results in:

$$\frac{(\sec\frac{\pi}{6} - 	an\frac{\pi}{6})(\sec\frac{\pi}{6} + 	an\frac{\pi}{6})}{\sec^2\frac{\pi}{6} - 	an^2\frac{\pi}{6}} = \frac{2}{\sqrt{3}}$$

However, $\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}$, so the correct value is:

$$\frac{1}{\cos\frac{\pi}{6}} = \frac{2}{\sqrt{3}} $$

Therefore, choice D is not equivalent.

Multiply; Then Use Fundamental Identities to Simplify the Expression Below