Question

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

Answer 1

To simplify the given expression ${\left( {\cos x + \sin x} \right)^2}$ which is in the form of ${\left( {a + b} \right)^2}$. So we can use the identity given by ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$ to simplify the given expression. As we know, ${\sin ^2}x + {\cos ^2}x = 1$, using this we will further simplify it. Then using the double angle formula of trigonometry i.e., $2\sin x\cos x = \sin 2x$ we find the result.

Complete step by step answer:
As see can that the given expression ${\left( {\cos x + \sin x} \right)^2}$ is similar to ${\left( {a + b} \right)^2}$ and we know that ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$.
Therefore, we can write the given expression as
$\Rightarrow {\left( {\cos x + \sin x} \right)^2} = {\cos ^2}x + 2\sin x\cos x + {\sin ^2}x$
On rewriting, we get
$\Rightarrow {\left( {\cos x + \sin x} \right)^2} = {\cos ^2}x + {\sin ^2}x + 2\sin x\cos x$
As we know from the trigonometric identities that ${\sin ^2}x + {\cos ^2}x = 1$. Therefore, we get
$\Rightarrow {\left( {\cos x + \sin x} \right)^2} = 1 + 2\sin x\cos x$
From the expression we can see that $2\sin x\cos x$ belongs to the trigonometric double angle identities. So, from a double angle identity we have $2\sin x\cos x = \sin 2x$.
So, we can replace $2\sin x\cos x$ with $\sin 2x$ in the above expression. So, on doing that we get
$\Rightarrow {\left( {\cos x + \sin x} \right)^2} = 1 + \sin 2x$
Therefore, the required answer or simplified form of ${\left( {\cos x + \sin x} \right)^2}$ is $1 + \sin 2x$.

Note:
Here, we have substituted $a = \cos x$ and $b = \sin x$ in ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$. This is because ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$ is an identity and an identity is an equation which is always true, no matter what values are substituted whereas an equation may not be true for some values that are substituted.