Question

Find the value of ${{\cos }^{4}}A-{{\sin }^{4}}A$ :
(a) sin2A
(b) –sin2A
(c) cos2A
(d) –cos2A

Answer 1

Hint: We are going to use some trigonometric formulas and some formulas of algebra also to simplify the given expression. First we will use the formula of $\left( {{a}^{2}}-{{b}^{2}} \right)$ to simplify it and then trigonometric formula to get the answer.

Complete step-by-step answer:
Let’s first write all the formulas that we are going use to solve this question,
$\begin{align}
  & \left( {{a}^{2}}-{{b}^{2}} \right)=\left( a-b \right)\left( a+b \right) \\
 & {{\cos }^{2}}x-{{\sin }^{2}}x=\cos 2x \\
 & {{\cos }^{2}}x+{{\sin }^{2}}x=1 \\
\end{align}$
Now let’s solve the question, first we will use $\left( {{a}^{2}}-{{b}^{2}} \right)$ this formula to simplify,
$\begin{align}
  & {{\cos }^{4}}A-{{\sin }^{4}}A \\
 & =\left( {{\cos }^{2}}A+{{\sin }^{2}}A \right)\left( {{\cos }^{2}}A-{{\sin }^{2}}A \right) \\
\end{align}$
Now using the above two trigonometric formulas we get,
$\left( \cos 2A \right)$
Hence the correct answer is cos2A.
Hence the correct option is (c).

Note: It’s always better that we check if the answer that we have got by using the above formula is correct or not to avoid some calculation mistake and for that we need to put some values in place of A and B to check whether it satisfies the above expression or not. There are a bunch of trigonometric formulas that should be kept in mind while solving these questions and if we use some different set of formulas then that will be another method to solve this question, but at some point we can see that they are nearly equal.