Question

Verify the following
$2\sin {{45}^{0}}\cos {{45}^{0}}=\sin {{90}^{0}}$

Answer 1

Hint: The Trigonometric ratios table helps to find the values of trigonometric standard angles such as ${{0}^{0}},{{30}^{0}},{{45}^{0}},{{60}^{0}}$ and ${{90}^{0}}$. It consists of trigonometric ratios – sine, cosine, tangent, cosecant, secant and cotangent. These ratios can be written in short as sin, cos, tan, cosec, sec and cot.

Complete step-by-step answer:
The value of the trigonometric ratios by using the trigonometric table is given below.

$\sin {{45}^{0}}=\dfrac{1}{\sqrt{2}},\cos {{45}^{0}}=\dfrac{1}{\sqrt{2}},\sin {{90}^{0}}=1$

Let us consider the left side of the given expression

\[2\sin {{45}^{0}}\cos {{45}^{0}}=2\times \dfrac{1}{\sqrt{2}}\times \dfrac{1}{\sqrt{2}}\]

Multiplying the terms on the right side, we get

\[2\sin {{45}^{0}}\cos {{45}^{0}}=2\times \dfrac{1}{2}\]

Cancelling the term 2 on the right side, we get

\[2\sin {{45}^{0}}\cos {{45}^{0}}=1\]

From the trigonometric table, $1=\sin {{90}^{0}}$

$2\sin {{45}^{0}}\cos {{45}^{0}}=\sin {{90}^{0}}$

Hence the given expression is verified


Note: Alternatively, the given question is verified by using the sine double angle formula that tells us that $\sin 2\theta $ is equal to $2\sin \theta \cos \theta $ that is $\sin 2A=2\sin A\cos A$. Let $A={{45}^{0}}$, then. $2\sin {{45}^{0}}\cos {{45}^{0}}=\sin \left( 2\times 45 \right)=\sin {{90}^{0}}$.