How to prove ${\sin ^2}\theta - {\cos ^2}\theta = 1 - 2{\cos ^2}\theta $

Hint:
Here, we will solve the LHS by using the relation between the sum of squares of sine and cosine. We will substitute the trigonometric identity and simplify it further to prove the given expression. Trigonometry is a branch of mathematics which helps us to study the relationship between the sides and the angles of a triangle.

Formula Used:
${\sin ^2}\theta + {\cos ^2}\theta = 1$

Complete step by step solution:
In order to prove that ${\sin ^2}\theta - {\cos ^2}\theta = 1 - 2{\cos ^2}\theta $, we will try solving the LHS to make it equal to the RHS.
Thus, we have,
LHS $ = {\sin ^2}\theta - {\cos ^2}\theta $……………………..$\left( 1 \right)$
Now, we know the trigonometric relation between the sum of square of sine and cosine.
This can also be written as ${\sin ^2}\theta + {\cos ^2}\theta = 1$
Or ${\sin ^2}\theta = 1 - {\cos ^2}\theta $
Hence, substituting this value in $\left( 1 \right)$, we get,
LHS \[ = 1 - {\cos ^2}\theta - {\cos ^2}\theta = 1 - 2{\cos ^2}\theta = \]RHS
Hence, LHS $ = $ RHS
Therefore,
${\sin ^2}\theta - {\cos ^2}\theta = 1 - 2{\cos ^2}\theta $
Hence, proved
Also, these formulas are interchangeable and if only cosine is being used in some questions, we can use the formula in RHS to solve that question further.

Note:
In this question, we have used trigonometry. In practical life, trigonometry is used by cartographers to make maps. It is also used by the aviation and naval industries. In fact, trigonometry is even used by Astronomers to find the distance between two stars. Hence, it has an important role to play in everyday life. The three most common trigonometric functions are the tangent function, the sine and the cosine function. In simple terms, they are written as ‘sin’, ‘cos’ and ‘tan’. Hence, trigonometry is not just a chapter to study, in fact, it is being used in everyday life.