Solve sin^2pi/8-sin^22pi/8

$$\sin ^ { 2 } \frac { \pi } { 8 } - \sin ^ { 2 } \frac { 2 \pi } { 8 }$$

Evaluate

$$-\frac{\sqrt{2}}{4}\approx -0.353553391$$

Short Solution Steps

Get the value of \sin(\frac{\pi }{8}) from trigonometric values table.

$$\left(\frac{\sqrt{2-\sqrt{2}}}{2}\right)^{2}-\left(\sin(\frac{2\pi }{8})\right)^{2}$$

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

$$\frac{\left(\sqrt{2-\sqrt{2}}\right)^{2}}{2^{2}}-\left(\sin(\frac{2\pi }{8})\right)^{2}$$

Divide 2\pi by 8 to get \frac{1}{4}\pi .

$$\frac{\left(\sqrt{2-\sqrt{2}}\right)^{2}}{2^{2}}-\left(\sin(\frac{1}{4}\pi )\right)^{2}$$

Get the value of \sin(\frac{1}{4}\pi ) from trigonometric values table.

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

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

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

The square of \sqrt{2} is 2.

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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}.

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

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

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

Calculate \sqrt{2-\sqrt{2}} to the power of 2 and get 2-\sqrt{2}.

$$\frac{2-\sqrt{2}}{2^{2}}-\frac{1}{2}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{2-\sqrt{2}}{4}-\frac{1}{2}$$

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

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

Since \frac{2-\sqrt{2}}{4} and \frac{2}{4} have the same denominator, subtract them by subtracting their numerators.

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

Do the calculations in 2-\sqrt{2}-2.

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