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.