Question

# Establish identity. 9\sec^2\theta-5\tan^2\theta=5+4\sec^2\theta

Trigonometric equation and identitie
Establish identity
$$\displaystyle{9}{{\sec}^{{2}}\theta}-{5}{{\tan}^{{2}}\theta}={5}+{4}{{\sec}^{{2}}\theta}$$

2021-09-10
Given identity $$\displaystyle{9}{{\sec}^{{2}}\theta}-{5}{{\tan}^{{2}}\theta}={5}+{4}{{\sec}^{{2}}\theta}$$
First from the left-hand side,
$$\displaystyle{9}{{\sec}^{{2}}\theta}-{5}{{\tan}^{{2}}\theta}$$
Here use the Pythagorean trigonometry identity, $$\displaystyle{{\tan}^{{2}}\theta}+{1}={{\sec}^{{2}}\theta}$$
$$\displaystyle{9}{{\sec}^{{2}}\theta}-{5}{{\tan}^{{2}}\theta}={9}{{\sec}^{{2}}\theta}-{5}{\left({{\sec}^{{2}}\theta}-{1}\right)}$$
$$\displaystyle={9}{{\sec}^{{2}}\theta}-{5}{{\sec}^{{2}}\theta}+{5}$$
$$\displaystyle={5}+{4}{{\sec}^{{2}}\theta}$$
Therefore it is established that the left-hand side equal to the right-hand side quantity.