Question

How do you simplify ${\sin ^4}x - {\cos ^4}x$ ?

Answer 1

Hint:
The given problem requires us to simplify the given trigonometric expression. The question requires thorough knowledge of trigonometric functions, formulae and identities. The question describes the wide ranging applications of trigonometric identities and formulae. We must keep in mind the trigonometric identities while solving such questions.

Complete step by step solution:
In the given question, we are required to evaluate the value of ${\sin ^4}x - {\cos ^4}x$ using the basic concepts of trigonometry and identities.
First we simplify the given trigonometric expression using algebraic identity $\left( {{a^2} - {b^2}} \right) = \left( {a - b} \right)\left( {a + b} \right)$. This identity simplifies the given trigonometric expression by factorising it as product of two factors.
So, we have, ${\sin ^4}x - {\cos ^4}x$.
$ \Rightarrow \left( {{{\sin }^2}x - {{\cos }^2}x} \right)\left( {{{\sin }^2}x + {{\cos }^2}x} \right)$
Now, using the trigonometric identity ${\sin ^2}\theta + {\cos ^2}\theta = 1$, we get
$ \Rightarrow \left( {{{\sin }^2}x - {{\cos }^2}x} \right)\left( 1 \right)$
$ \Rightarrow \left( {{{\sin }^2}x - {{\cos }^2}x} \right)$
So, we have simplified the trigonometric expression a bit but it can be further simplified by using the double angle formula for cosine.
Now, we know that $\cos \left( {2\theta } \right) = {\cos ^2}\theta - {\sin ^2}\theta $.
$ \Rightarrow - \left( {{{\cos }^2}x - {{\sin }^2}x} \right)$
$ \Rightarrow - \cos \left( {2x} \right)$

So, we get the value of the trigonometric expression ${\sin ^4}x - {\cos ^4}x$ as $ - \cos \left( {2x} \right)$ .

Note:
There are six trigonometric ratios: $\sin \theta $, $\cos \theta $, $\tan \theta $, $\cos ec\theta $, $\sec \theta $and $\cot \theta $. Basic trigonometric identities include ${\sin ^2}\theta + {\cos ^2}\theta = 1$, ${\sec ^2}\theta = {\tan ^2}\theta + 1$ and $\cos e{c^2}\theta = {\cot ^2}\theta + 1$. These identities are of vital importance for solving any question involving trigonometric functions and identities. All the trigonometric ratios can be converted into each other using the simple trigonometric identities listed above.The given problem involves the use of trigonometric formulae and identities. Such questions require thorough knowledge of trigonometric conversions and ratios. Algebraic operations and rules like transposition rule come into significant use while solving such problems.