Question

How do you prove $\cos 2\theta = {\cos ^4}\theta - {\sin ^4}\theta $ ?

Answer 1

Hint: The given question belongs to the solution of the trigonometric equation. To solve the given equation we will use the trigonometric ratio identities. Here we are going to relate the given trigonometric equation to the algebraic equation to solve it easily. We will use the double angle formula of the trigonometric ratio to find the solution of the given trigonometric equation. We know that the trigonometric ratio $\cos 2\theta $ have different formula as,
$\cos 2\theta = {\cos ^2}\theta - {\sin ^2}\theta $
$\cos 2\theta = 1 - 2{\sin ^2}\theta $

Complete step by step answer:
Step: 1 the given trigonometric equation is,
$\cos 2\theta = {\cos ^4}\theta - {\sin ^4}\theta $
Now consider the right hand side of the equation,
$ \Rightarrow {\cos ^4}\theta - {\sin ^4}\theta $
We can expand the given trigonometric equation by using the algebraic formula.
The basic algebraic formula is,
${a^4} - {b^4} = \left( {{a^2} - {b^2}} \right)\left( {{a^2} + {b^2}} \right)$
 Step: 2 now compare the right hand side of the given trigonometric equation with the formula.
Therefore we can write the left hand side of the given trigonometric equation in the form of an algebraic formula.
So,
$ \Rightarrow {\cos ^4}\theta - {\sin ^4}\theta = \left( {{{\cos }^2}\theta - {{\sin }^2}\theta } \right)\left( {{{\cos }^2}\theta + {{\sin }^2}\theta } \right)$
 Step: 3 now use the trigonometric identities formula to solve the given trigonometric equation.
We know that the trigonometric identities formula is,
${\cos ^2}\theta + {\sin ^2}\theta = 1$
Substitute ${\cos ^2}\theta + {\sin ^2}\theta = 1$ in the given trigonometric equation to solve the equation.
$ \Rightarrow {\cos ^4}\theta - {\sin ^4}\theta = \left( {{{\cos }^2}\theta - {{\sin }^2}\theta } \right) \times 1$
Now simplify the equation to its simplest form.
$ \Rightarrow {\cos ^4}\theta - {\sin ^4}\theta = \left( {{{\cos }^2}\theta - {{\sin }^2}\theta } \right)$
 Step: 4 now use double angle formula of trigonometric ratios to solve the equation.
We know that the double angle formula of trigonometric ratio is,
$\cos 2\theta = {\cos ^2}\theta - {\sin ^2}\theta $
 Substitute the value of $\cos 2\theta = {\cos ^2}\theta - {\sin ^2}\theta = \cos 2\theta $ in the equation to find the solution of the equation.
$ \Rightarrow {\cos ^4}\theta - {\sin ^4}\theta = \cos 2\theta $

Note: Students are advised to use the double angle formula of the trigonometric ratios. They must know the standard formula of trigonometric ratios, ${\cos ^2}\theta + {\sin ^2}\theta = 1$ .They should not make mistakes while solving the equation. They must remember the algebraic formula ${a^4} - {b^4} = \left( {{a^2} - {b^2}} \right)\left( {{a^2} + {b^2}} \right)$.