Question

How do you prove ${\left( {\sin x + \cos x} \right)^2} = 1 + 2\sin x\cos x$ ?

Answer 1

Hint : In this question we need to prove ${\left( {\sin x + \cos x} \right)^2} = 1 + 2\sin x\cos x$ .
In order to prove we will consider the LHS and evaluate it using the formula of ${\left( {a + b} \right)^2}$ . Then, we will apply the trigonometric formula and evaluate it to determine the required proof.

Complete step-by-step answer :
Here, we will prove ${\left( {\sin x + \cos x} \right)^2} = 1 + 2\sin x\cos x$
Now let us consider the LHS,
LHS $= {\left( {\sin x + \cos x} \right)^2}$
So, we can see that ${\left( {\sin x + \cos x} \right)^2}$ is in the form of ${\left( {a + b} \right)^2}$ .
Now, we know that ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$
Here, $a = \sin x$ and $b = \cos x$
Thus, we will substitute the value of $a$ and $b$ in the formula, we have,
${\left( {\sin x + \cos x} \right)^2} = {\sin ^2}x + 2\sin x\cos x + {\cos ^2}x$
Now, let us reorder the terms,
${\left( {\sin x + \cos x} \right)^2} = {\sin ^2}x + {\cos ^2}x + 2\sin x\cos x$
From trigonometric identities we know that ${\sin ^2}x + {\cos ^2}x = 1$ .
Let us apply the value here, we have,
${\left( {\sin x + \cos x} \right)^2} = 1 + 2\sin x\cos x$
Therefore, LHS=RHS
Hence proved.
So, the correct answer is “ ${\left( {\sin x + \cos x} \right)^2} = 1 + 2\sin x\cos x$ ”.

Note : In this question it is important to note that whenever we come across these kinds of questions, we always start from the more complex side. This is because it is a lot easier to eliminate terms to make a complex function simple than to find ways to introduce terms to make a simple function complex. Take one step, watch one step.