How do you prove \[\cos 2x = 1 - 2{\sin ^2}x\] using other trigonometric identities?

Hint:Here, we will first consider the left-hand side of the equation. Then we will use the half-angle formula and also use the trigonometric identities of the relationship between the squares of sine and cosine. Simplifying the equation further we will prove both sides are equal and thus, prove the given equation.

Formula Used:
We will use the following formulas:
\[\cos 2x = {\cos ^2}x - {\sin ^2}x\]
\[{\cos ^2}x + {\sin ^2}x = 1\]

Complete step-by-step answer:
To prove: \[\cos 2x = 1 - 2{\sin ^2}x\]
Proof: Here, we will first consider the left-hand side of the given equation.
LHS \[ = \cos 2x\]
Now, using half angle formulas, we know that,
\[ \Rightarrow \cos 2x = {\cos ^2}x - {\sin ^2}x\]………………………….\[\left( 1 \right)\]
Also, we know that, \[{\cos ^2}x + {\sin ^2}x = 1\]
\[ \Rightarrow {\cos ^2}x = 1 - {\sin ^2}x\]…………………………………\[\left( 2 \right)\]
Substituting equation \[\left( 2 \right)\] in equation \[\left( 1 \right)\], we get,
\[\cos 2x = \left( {1 - {{\sin }^2}x} \right) - {\sin ^2}x\]
Adding the like terms, we get
\[ \Rightarrow \cos 2x = 1 - 2{\sin ^2}x = \] RHS
Therefore,
LHS \[ = \] RHS
Hence, \[\cos 2x = 1 - 2{\sin ^2}x\]
Hence, proved

Note: In this question, we have used trigonometry identity to prove this question. Trigonometry is a branch of mathematics that helps us to study the relationship between the sides and the angles of a triangle. 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.