Question

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

Trigonometric equation and identitie
ANSWERED
asked 2021-09-09
Establish identity
\(\displaystyle{9}{{\sec}^{{2}}\theta}-{5}{{\tan}^{{2}}\theta}={5}+{4}{{\sec}^{{2}}\theta}\)

Expert Answers (1)

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.
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