Question

How do you prove \[\cos 2x = {\cos ^2}x - {\sin ^2}x\] using other trigonometric identities?

Answer 1

Hint: Here we are given a trigonometric identity that is \[\cos 2x = {\cos ^2}x - {\sin ^2}x\] and we are asked to prove this identity using other trigonometric identities. For approaching this question we need to know a basic trigonometric identity that is as follows \[\cos (\alpha + \beta ) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \] applying this and further simplification we could get through the asked proof easily.

Complete step-by-step answer:
Here we are given an equation that is \[\cos 2x = {\cos ^2}x - {\sin ^2}x\] and we are asked the method to proof that can be done as taking the LHS of the given equation that is \[\cos 2x\] and applying the trigonometric identity that is \[\cos (\alpha + \beta ) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \] on it and further simplifying it that is depicted as follows –
Taking the LHS of the given equation \[\cos 2x = {\cos ^2}x - {\sin ^2}x\]that is \[\cos 2x\] and applying the identity that is \[\cos (\alpha + \beta ) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \]-
Here \[\cos 2x\] can be expressed as -
\[\cos 2x = \cos (x + x)\]
Now applying the identity as stated above that is of \[\cos (\alpha + \beta ) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \]
Here the \[\alpha = \beta = x\] so the RHS becomes –
\[\cos x\cos x - \sin x\sin x\]
Now further simplifying the resultant as multiplying the like terms originated that comes out to be as –
\[{\cos ^2}x - {\sin ^2}x\]
Which is equal to the RHS of the required proof that is \[\cos 2x = {\cos ^2}x - {\sin ^2}x\]
Therefore LHS=RHS hence proved the required quantity that is -\[\cos 2x = {\cos ^2}x - {\sin ^2}x\].

Note: While solving such kind of the questions one should know the identities of the form \[\cos (\alpha + \beta ) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \],\[\sin (\alpha + \beta ) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \] which is used to express the angles and their trigonometric expressions in the desired way which would help to get the desired answer.