Question

Evaluating an Expression

Evaluate the expression without using a calculator

${\left(cos{30}^{°}\right)}^{\mathbf{2}}\mathbf{-}{\left(sin{30}^{°}\right)}^{\mathbf{2}}$

Short answer

The answer of the expression is $\frac{\mathbf{1}}{\mathbf{2}}$

Step-by-step solution

Step 1. Given information

A trigonometric expression is given as ${\left(\cos {30}^{°}\right)}^2-{\left(\sin {30}^{°}\right)}^2$.

Step 2. Concept used

Use trigonometric ratios to make the problem easier to solve. Make the necessary changes to the ratio values in the expression and perform the computation. Use trigonometric identities to simplify things even further.

Step 3. Calculation

The given expression is ${\left(\cos {30}^{°}\right)}^2-{\left(\sin {30}^{°}\right)}^2$, substitute the values as-

$\begin{array}{l}\because \cos {30}^{°}=\frac{\sqrt{3}}{2}\&\sin {30}^{°}=\frac{1}{2}\\ ={\left(\frac{\sqrt{3}}{2}\right)}^2-{\left(\frac{1}{2}\right)}^2\\ =\frac{3}{4}-\frac{1}{4}\\ =\frac{2}{4}\\ =\frac{1}{2}\end{array}$