Question

How do you prove trigonometric identity $1 + {\tan ^2}x = {\sec ^2}x$?

Answer 1

This question is from the topic of trigonometric formulas. In this question we have to prove trigonometric identity $1 + {\tan ^2}x = {\sec ^2}x$. To prove this we will use the definition of tangent and secant trigonometric functions and trigonometric formula ${\sin ^2}x + {\cos ^2}x = 1$. We prove this by starting with L.H.S and getting R.H.S by using properties or formulas.

Complete step by step solution:
Let us try to solve the question in which we are asked to prove $1 + {\tan ^2}x = {\sec ^2}x$. To solve these types of questions we need to know the definition and trigonometric formulas without them it is impossible to solve these questions. To prove this we will first use some of this property.
Here are those properties of trigonometric functions:
3) $\tan x = \dfrac{{\sin x}}{{\cos x}}$
4) $\sec x = \dfrac{1}{{\cos x}}$
And use the trigonometric formula ${\sin ^2}x + {\cos ^2}x = 1$to prove the given trigonometric formula $1 + {\tan ^2}x = {\sec ^2}x$.
To prove: $1 + {\tan ^2}x = {\sec ^2}x$
Proof: To prove the above equation we will start with the L.H.S of equation.
So we have,
$L.H.S = 1 + {\tan ^2}x$ $eq(1)$
As we know that $\tan x = \dfrac{{\sin x}}{{\cos x}}$, so we have ${\tan ^2}x = \dfrac{{{{\sin }^2}x}}{{{{\cos }^2}x}}$ putting this back into $eq(1)$, we get
$L.H.S = 1 + \dfrac{{{{\sin }^2}x}}{{{{\cos }^2}x}}$
Now performing fraction addition we get,
$L.H.S = 1 + \dfrac{{{{\sin }^2}x}}{{{{\cos }^2}x}} \\\ L.H.S\,\, = \dfrac{{{{\cos }^2}x + {{\sin }^2}x}}{{{{\cos }^2}x}} \\\$
Now using the trigonometric formula ${\sin ^2}x + {\cos ^2}x = 1$. We get,
$L.H.S = \dfrac{1}{{{{\cos }^2}x}}$
Now using property 2 from above which is $\sec x = \dfrac{1}{{\cos x}}$. We get,
$L.H.S = {\sec ^2}x$
$1 + {\tan ^2}x = L.H.S = R.H.S = {\sec ^2}x$
Hence we have proved that$1 + {\tan ^2}x = {\sec ^2}x$.

Note: Generally questions in which we are asked to prove a trigonometric formula from L.H.S of equation but this can be proved also starting with R.H.S. In proving the above formula we just have to apply reverse steps of the above proof. You can try to prove it by starting with ${\sec ^2}x$. For solving this type of question you are required to know the trigonometric formulas.