Choose the correct option. Justify your answer $9{\sec ^2}A - 9{\tan ^2}A = $ A) 1 B) 9 C) 8 D) 0
Answer
Verified
Hint: Here, we will solve the given trigonometric equation by the use of the trigonometric identity ${\sec ^2}A - {\tan ^2}A = 1$ and trigonometric formulae. Given, $9{\sec ^2}A - 9{\tan ^2}A \to (1)$ Let us common $9$ from equation (1), we get $ \Rightarrow 9({\sec ^2}A - {\tan ^2}A) \to (2)$ Since, we know the trigonometric identity i.e.., ${\sec ^2}A - {\tan ^2}A = 1$. Therefore, equation (2) can be written as $ \Rightarrow 9(1) = 9$ Hence, the value of $9{\sec ^2}A - 9{\tan ^2}A$ is $9$. Therefore, from the given options ‘B’ is correct. Note: The elaborate approach for the given equation is by substituting $\sec A$ with $\dfrac{1}{{\cos A}}$ and $\tan A$ with $\dfrac{{\sin A}}{{\cos A}}$ .
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