Question

The value of ${\cos ^4}\theta - {\sin ^4}\theta$ is equal to:
A. $1 + 2{\sin ^2}\left( {\dfrac{\theta }{2}} \right)$
B. $2{\cos ^2}\left( \theta \right) - 1$
C. $1 - 2{\sin ^2}\left( {\dfrac{\theta }{2}} \right)$
D. $2{\cos ^2}\left( \theta \right) + 1$

Answer 1

The given question deals with finding the value of trigonometric expression doing basic simplification of trigonometric functions by using some of the simple trigonometric formulae such as ${\cos ^2}x + {\sin ^2}x = 1$ and double angle formula for cosine. Basic algebraic rules and trigonometric identities are to be kept in mind while doing simplification in the given problem. We must know the simplification rules to solve the problem with ease.

Complete step by step answer:
In the given problem, we have to find the value of ${\cos ^4}\theta - {\sin ^4}\theta$.
So, we have, ${\cos ^4}\theta - {\sin ^4}\theta$
Using the algebraic identity $\left( {{a^2} - {b^2}} \right) = \left( {a - b} \right)\left( {a + b} \right)$, we get,
$\Rightarrow {\left( {{{\cos }^2}\theta } \right)^2} - {\left( {{{\sin }^2}\theta } \right)^2}$
$\Rightarrow \left( {{{\cos }^2}\theta - {{\sin }^2}\theta } \right)\left( {{{\cos }^2}\theta + {{\sin }^2}\theta } \right)$
Now, we subtract the trigonometric identity ${\cos ^2}x + {\sin ^2}x = 1$ in the above equation. So, we get,
$\Rightarrow \left( {{{\cos }^2}\theta - {{\sin }^2}\theta } \right)\left( 1 \right)$
Simplifying the expression, we get,
$\Rightarrow {\cos ^2}\theta - {\sin ^2}\theta$
Now, we know the double angle formula of cosine as $\cos 2x = {\cos ^2}x - {\sin ^2}x$. So, we get,
$\Rightarrow \cos \left( {2\theta } \right)$
We also know the double angle formula for cosine as $\cos 2x = 2{\cos ^2}x - 1$. Applying this in the expression, we get,
$\Rightarrow 2{\cos ^2}\theta - 1$
Therefore, ${\cos ^4}\theta - {\sin ^4}\theta = 2{\cos ^2}\theta - 1$ is the required result.

Hence, option B is the correct answer.

Note: We must have a strong grip over the concepts of trigonometry, related formulae and rules to ace these types of questions. Besides these simple trigonometric formulae, trigonometric identities are also of significant use in such types of questions where we have to simplify trigonometric expressions with help of basic knowledge of algebraic rules and operations. However, questions involving this type of simplification of trigonometric ratios may also have multiple interconvertible answers but we have to mark the most appropriate option among the given choices.