Solveeit Logo
Login

Question

Question: How do you simplify \( {\tan ^2}\phi - {\sec ^2}\phi \) ?...

How do you simplify tan⁡2ϕ−sec⁡2ϕ{\tan ^2}\phi - {\sec ^2}\phi ?

Explanation

Solution

Hint : In the given question, we are given the difference of two trigonometric functions, the trigonometric ratios are raised to the power 2, that is, we are given the difference of the squares of two trigonometric ratios and we have to simplify them. The simplest trigonometric ratios are the sine and cosine functions, so we convert the given ratios in the terms of sine and cosine ratios. Then we use an identity that the sum of the squares of sine and cosine is equal to 1. This way we can simplify the given expression.

Complete step-by-step answer :
We know that –
tan⁡ϕ=sin⁡ϕcos⁡ϕ ⇒tan⁡2ϕ=sin⁡2ϕcos⁡2ϕ  \tan \phi = \dfrac{{\sin \phi }}{{\cos \phi }} \\\ \Rightarrow {\tan ^2}\phi = \dfrac{{{{\sin }^2}\phi }}{{{{\cos }^2}\phi }} \\\
And
sec⁡ϕ=1cos⁡ϕ ⇒sec⁡2ϕ=1cos⁡2ϕ   \sec \phi = \dfrac{1}{{\cos \phi }} \\\ \Rightarrow {\sec ^2}\phi = \dfrac{1}{{{{\cos }^2}\phi }} \;
Using these two values in the given equation, we get –
tan⁡2ϕ−sec⁡2ϕ=sin⁡2ϕcos⁡2ϕ−1cos⁡2ϕ=sin⁡2ϕ−1cos⁡2ϕ{\tan ^2}\phi - {\sec ^2}\phi = \dfrac{{{{\sin }^2}\phi }}{{{{\cos }^2}\phi }} - \dfrac{1}{{{{\cos }^2}\phi }} = \dfrac{{{{\sin }^2}\phi - 1}}{{{{\cos }^2}\phi }}
We know that,
sin⁡2ϕ+cos⁡2ϕ=1 ⇒sin⁡2ϕ−1=−cos⁡2ϕ   {\sin ^2}\phi + {\cos ^2}\phi = 1 \\\ \Rightarrow {\sin ^2}\phi - 1 = - {\cos ^2}\phi \;
Put this value, in the obtained expression –
tan⁡2ϕ−sec⁡2ϕ=−cos⁡2ϕcos⁡2ϕ ⇒tan⁡2ϕ−sec⁡2ϕ=−1   {\tan ^2}\phi - {\sec ^2}\phi = \dfrac{{ - {{\cos }^2}\phi }}{{{{\cos }^2}\phi }} \\\ \Rightarrow {\tan ^2}\phi - {\sec ^2}\phi = - 1 \;
Hence the simplified form of tan⁡2ϕ−sec⁡2ϕ{\tan ^2}\phi - {\sec ^2}\phi is -1.
So, the correct answer is “-1”.

Note : The obtained relation is used as a trigonometric identity that the difference of the squares of the secant function and the tangent function is one. Using this identity, we can find one unknown quantity when the other is known. When we replace the given values with other simpler values until it cannot be done further is known as the simplification of the expression. We use trigonometric identities to solve such types of questions like we used an identity of sine and cosine in this solution. At last, we obtained the value in numerical form, a single-digit integer, that is, -1, so it cannot be simplified further and is thus the required answer.