Question

In problems 19-98, establish each identity.

$9{\sec }^2\left(\theta \right)-5{\tan }^2\left(\theta \right)=5+4{\sec }^2\left(\theta \right)$

Short answer

We have proved the identity$9{\sec }^2\left(\theta \right)-5{\tan }^2\left(\theta \right)=5+4{\sec }^2\left(\theta \right)$.

Step-by-step solution

Step 1. Given information.

We have:

$9{\sec }^2\left(\theta \right)-5{\tan }^2\left(\theta \right)=5+4{\sec }^2\left(\theta \right)$

Step 2. Establish the identity.

Take the left hand side:

$9{\sec }^2\left(\theta \right)-5{\tan }^2\left(\theta \right)$

Apply identity: role="math" localid="1646583478228" ${\tan }^2\theta ={\sec }^2\left(\mathrm{\theta }\right)-1$
$$\begin{gathered} =9{\sec }^2\left(\theta \right)-5\left(se{c}^2\theta -1\right) \\ =9{\sec }^2\left(\theta \right)-5se{c}^2\theta +5 \\ =5+4se{c}^2\left(\theta \right) \end{gathered}$$

Hence, proved.