Solve {left(cos(45^circ)right)}^2-1/2tan(45^circ)+tan(30^circ)

$${ \left( \cos ( 45 ^ { \circ } ) \right) }^{ 2 } - \frac{ 1 }{ 2 } \tan ( 45 ^ { \circ } ) + \tan ( 30 ^ { \circ } )$$

Evaluate

$$\frac{\sqrt{3}}{3}\approx 0.577350269$$

Short Solution Steps

Get the value of \cos(45) from trigonometric values table.

$$\left(\frac{\sqrt{2}}{2}\right)^{2}-\frac{1}{2}\tan(45)+\tan(30)$$

To raise \frac{\sqrt{2}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}-\frac{1}{2}\tan(45)+\tan(30)$$

Get the value of \tan(45) from trigonometric values table.

$$\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}-\frac{1}{2}\times 1+\tan(30)$$

Multiply \frac{1}{2} and 1 to get \frac{1}{2}.

$$\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}-\frac{1}{2}+\tan(30)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2^{2} and 2 is 4. Multiply \frac{1}{2} times \frac{2}{2}.

$$\frac{\left(\sqrt{2}\right)^{2}}{4}-\frac{2}{4}+\tan(30)$$

Since \frac{\left(\sqrt{2}\right)^{2}}{4} and \frac{2}{4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(\sqrt{2}\right)^{2}-2}{4}+\tan(30)$$

Get the value of \tan(30) from trigonometric values table.

$$\frac{\left(\sqrt{2}\right)^{2}-2}{4}+\frac{\sqrt{3}}{3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4 and 3 is 12. Multiply \frac{\left(\sqrt{2}\right)^{2}-2}{4} times \frac{3}{3}. Multiply \frac{\sqrt{3}}{3} times \frac{4}{4}.

$$\frac{3\left(\left(\sqrt{2}\right)^{2}-2\right)}{12}+\frac{4\sqrt{3}}{12}$$

Since \frac{3\left(\left(\sqrt{2}\right)^{2}-2\right)}{12} and \frac{4\sqrt{3}}{12} have the same denominator, add them by adding their numerators.

$$\frac{3\left(\left(\sqrt{2}\right)^{2}-2\right)+4\sqrt{3}}{12}$$

The square of \sqrt{2} is 2.

$$\frac{2-2}{4}+\frac{\sqrt{3}}{3}$$

Subtract 2 from 2 to get 0.

$$\frac{0}{4}+\frac{\sqrt{3}}{3}$$

Zero divided by any non-zero number gives zero.

$$0+\frac{\sqrt{3}}{3}$$

Anything plus zero gives itself.

$$\frac{\sqrt{3}}{3}$$