Question

How do you rewrite (sin x – cos x) (sin x + cos x)?

Answer 1

Hint: We will first use the formula given by ${a^2} - {b^2} = (a - b)(a + b)$ and obtain the solution. Now, we will use the fact that ${\cos ^2}x - {\sin ^2}x = \cos 2x$ and thus, we have the required answer.

Complete Step by Step Solution:
We are given that we are required to re – write (sin x – cos x) (sin x + cos x).
We know that we have an identity given by the following expression with us:-
$ \Rightarrow {a^2} - {b^2} = (a - b)(a + b)$
Replacing a by sin x and b by cos x in the above mentioned formula, we will then obtain the following expression with us:-
$ \Rightarrow {\sin ^2}x - {\cos ^2}x = (\sin x + \cos x)(\sin x - \cos x)$ ……………….(1)
Now, we also know that we have a formula which says that ${\cos ^2}x - {\sin ^2}x = \cos 2x$.
Multiplying the above formula by a negative sign, we will then obtain the following expression with us:-
$ \Rightarrow {\sin ^2}x - {\cos ^2}x = - \cos 2x$
Putting this in equation (1) and re – arranging the left hand side and the right hand side, we will then obtain the following expression with us:-

$ \Rightarrow (\sin x - \cos x)(\sin x + \cos x) = - \cos 2x$
Thus, we have the required answer.

Note:
The students must note that we may also verify ${\sin ^2}x - {\cos ^2}x = (\sin x + \cos x)(\sin x - \cos x)$ instead of using the formula ${a^2} - {b^2} = (a - b)(a + b)$ as mentioned above. Let us solve it without the formula:-
The R H S of the equation is $(\sin x + \cos x)(\sin x - \cos x)$.
This is the expression which we are required to rewrite.
Let us first use the fact that (a + b) (c + d) = a(c + d) + b(c + d).
Therefore, we have: $\sin x(\sin x - \cos x) + \cos x(\sin x - \cos x)$.
Simplifying it further, we then obtain the following expression with us:-
$ \Rightarrow {\sin ^2}x - \sin x\cos x + \sin x\cos x - {\cos ^2}x$
Clubbing the like terms, we will then obtain the following expression with us:-
$ \Rightarrow {\sin ^2}x - {\cos ^2}x$
Thus, we have: L H S = R H S.
Hence, we have the formula given by the following expression with us as we used in the above solution:-
$ \Rightarrow {\sin ^2}x - {\cos ^2}x = (\sin x + \cos x)(\sin x - \cos x)$
Now, we can do the same with the rest as done in the solution.