Question

Find the value of \[9{\sec ^2}A - 9{\tan ^2}A\].

Answer 1

Hint: In this particular question use the trigonometric identities like \[{\sec ^2}x - {\tan ^2}x = 1\]. And then solve the equation to reach the solution of the question.

Complete step-by-step answer:
Now as we know that according to trigonometric identities \[{\sec ^2}x - {\tan ^2}x = 1\].
So, \[{\sec ^2}x - {\tan ^2}x = 1\] (2)
So, now let the value of the given equation be equal to y.
So, y = \[9{\sec ^2}A - 9{\tan ^2}A\] (2)
Now taking 9 common from the Right-hand side (RHS) of the above equation.
\[ \Rightarrow y = 9\left( {{{\sec }^2}A - {{\tan }^2}A} \right)\]
Now putting the value of equation 1 in equation 2 to solve the above equation.
 \[ \Rightarrow y = 9\left( 1 \right) = 9\]
So, as stated above in the solution that y = \[9{\sec ^2}A - 9{\tan ^2}A\].
Hence, \[9{\sec ^2}A - 9{\tan ^2}A\] = 9

Note:Whenever we face such types of questions the key concept we have to remember is that we always recall the trigonometric identities of the trigonometric functions used in the given equation. Use identities like \[{\sec ^2}x - {\tan ^2}x = 1\], \[{\sin ^2}x + {\cos ^2}x = 1\] and \[co{\sec ^2}x - co{t^2}x = 1\]. And then simplify the given equation and we will get the required answer.