Question

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

Answer 1

The given question deals with finding the value of trigonometric expression by first expanding the whole square term using the algebraic identity ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$ and then using the trigonometric identity ${\cos ^2}x + {\sin ^2}x = 1$ to simplify the trigonometric expression. Basic algebraic rules and trigonometric identities are to be kept in mind while doing simplification in the given problem. We must know the simplification rules to solve the problem with ease.

Complete step by step answer:
In the given problem, we have to simplify the trigonometric expression: ${\left( {\sin \theta + \cos \theta } \right)^2}$.
So, we have, ${\left( {\sin \theta + \cos \theta } \right)^2}$
Using the algebraic identity ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$, we get,
$\Rightarrow \left( {{{\sin }^2}\theta + 2\sin \theta \cos \theta + {{\cos }^2}\theta } \right)$
Grouping the square terms together to use resemble the trigonometric identity ${\cos ^2}x + {\sin ^2}x = 1$, we get,
$\Rightarrow \left( {{{\sin }^2}\theta + {{\cos }^2}\theta } \right) + 2\sin \theta \cos \theta$
Using the trigonometric identity ${\cos ^2}x + {\sin ^2}x = 1$ in the problem, we have,
$\Rightarrow 1 + 2\sin \theta \cos \theta$
Now, we know the double angle formula of sine as $\sin 2x = 2\sin x\cos x$. So, we get,
$\Rightarrow 1 + \sin \left( {2\theta } \right)$

Therefore, the simplified form of ${\left( {\sin \theta + \cos \theta } \right)^2}$ is $1 + \sin \left( {2\theta } \right)$.

Note: The trigonometric identities ${\cos ^2}x + {\sin ^2}x = 1$ makes a crucial step in simplifying the given trigonometric expression. The other two trigonometric identities: ${\sec ^2}x = 1 + {\tan ^2}x$ and $\cos e{c^2}x = 1 + {\cot ^2}x$ also have their significance in solving such problems. The double angle formula of sine $\sin 2x = 2\sin x\cos x$ makes the final result more simplified. Otherwise, if we leave the answer at $1 + 2\sin \theta \cos \theta$, it will also be correct. The algebraic identities such as ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$ help in simplification of the expression and evaluating the whole square of a binomial involving addition of two terms.