How do you verify the identity: ${\cos ^2}x - {\sin ^2}x = 1 - 2{\sin ^2}x?$

Hint: Here we will verify the given expression using the third identity which states that ${\sin ^2}x + {\cos ^2}x = 1$Also, here will take both the sides of the equation one by one and then verify it.

Complete step-by-step solution:
Now take the left hand side of the given equation,
LHS $ = {\cos ^2}x - {\sin ^2}x$
Now, using the identity ${\sin ^2}x + {\cos ^2}x = 1$will find the value of the cosine function and place it in the LHS.
When any term is moved from one side to another, then the sign of the term also changes. Positive terms become negative and vice-versa.
${\cos ^2}x = 1 - {\sin ^2}x$
LHS $ = 1 - {\sin ^2}x - {\sin ^2}x$
While simplification between the like terms always remember that, when there are two minus signs, you have to perform addition but apply negative sign to the resultant value.
LHS $ = 1 - 2{\sin ^2}x$
By observing the given expression, we can say that the value of LHS is the same as RHS.
Therefore,
$LHS = RHS$
Hence, proved.

Additional Information: Cosine and secant are inverse function of each other and sine and cosecant are the inverse functions of each other.

Note: Always remember the basic trigonometric identities for the accurate and an efficient solution. Now the correlation between them. While doing simplification remember the golden rules-
i) Addition of two positive terms gives the positive term
ii) Addition of one negative and positive term, you have to do subtraction and give signs of bigger numbers whether positive or negative.
iii) Addition of two negative numbers gives a negative number but in actual you have to add both the numbers and give a negative sign to the resultant answer.