Solve {left(cos(pi/4)right)}^2+{left(cos(pi/3)right)}^2+{left(cos(pi/3)right)}^2

$${ \left( \cos ( \frac{ \pi }{ 4 } ) \right) }^{ 2 } + { \left( \cos ( \frac{ \pi }{ 3 } ) \right) }^{ 2 } + { \left( \cos ( \frac{ \pi }{ 3 } ) \right) }^{ 2 }$$

Evaluate

$$1$$

Solution Steps

Get the value of \cos(\frac{\pi }{4}) from trigonometric values table.

$$\left(\frac{\sqrt{2}}{2}\right)^{2}+\left(\cos(\frac{\pi }{3})\right)^{2}+\left(\cos(\frac{\pi }{3})\right)^{2}$$

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

$$\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}+\left(\cos(\frac{\pi }{3})\right)^{2}+\left(\cos(\frac{\pi }{3})\right)^{2}$$

Get the value of \cos(\frac{\pi }{3}) from trigonometric values table.

$$\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}+\left(\frac{1}{2}\right)^{2}+\left(\cos(\frac{\pi }{3})\right)^{2}$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}+\frac{1}{4}+\left(\cos(\frac{\pi }{3})\right)^{2}$$

Get the value of \cos(\frac{\pi }{3}) from trigonometric values table.

$$\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}+\frac{1}{4}+\left(\frac{1}{2}\right)^{2}$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}+\frac{1}{4}+\frac{1}{4}$$

Add \frac{1}{4} and \frac{1}{4} to get \frac{1}{2}.

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

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}.

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

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

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

The square of \sqrt{2} is 2.

$$\frac{2}{2^{2}}+\frac{1}{2}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{2}{4}+\frac{1}{2}$$

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

$$\frac{1}{2}+\frac{1}{2}$$

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

$$1$$

Factor

$$1$$