Solve 2sin^{2}45^circ+cos^245^circ-tan60^circ

$$2 \sin ^ { 2 } 45 ^ { \circ } + \cos ^ { 2 } 45 ^ { \circ } - \tan 60 ^ { \circ }$$

Evaluate

$$\frac{3}{2}-\sqrt{3}\approx -0.232050808$$

Solution Steps

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

$$2\times \left(\frac{\sqrt{2}}{2}\right)^{2}+\left(\cos(45)\right)^{2}-\tan(60)$$

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

$$2\times \frac{\left(\sqrt{2}\right)^{2}}{2^{2}}+\left(\cos(45)\right)^{2}-\tan(60)$$

Express 2\times \frac{\left(\sqrt{2}\right)^{2}}{2^{2}} as a single fraction.

$$\frac{2\left(\sqrt{2}\right)^{2}}{2^{2}}+\left(\cos(45)\right)^{2}-\tan(60)$$

Cancel out 2 in both numerator and denominator.

$$\frac{\left(\sqrt{2}\right)^{2}}{2}+\left(\cos(45)\right)^{2}-\tan(60)$$

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

$$\frac{\left(\sqrt{2}\right)^{2}}{2}+\left(\frac{\sqrt{2}}{2}\right)^{2}-\tan(60)$$

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

$$\frac{\left(\sqrt{2}\right)^{2}}{2}+\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}-\tan(60)$$

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

$$\frac{2\left(\sqrt{2}\right)^{2}}{4}+\frac{\left(\sqrt{2}\right)^{2}}{4}-\tan(60)$$

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

$$\frac{2\left(\sqrt{2}\right)^{2}+\left(\sqrt{2}\right)^{2}}{4}-\tan(60)$$

Get the value of \tan(60) from trigonometric values table.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply \sqrt{3} times \frac{4}{4}.

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

Since \frac{2\left(\sqrt{2}\right)^{2}+\left(\sqrt{2}\right)^{2}}{4} and \frac{4\sqrt{3}}{4} have the same denominator, subtract them by subtracting their numerators.

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

The square of \sqrt{2} is 2.

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

Multiply 2 and 2 to get 4.

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

The square of \sqrt{2} is 2.

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

Add 4 and 2 to get 6.

$$\frac{6}{4}-\sqrt{3}$$

Reduce the fraction \frac{6}{4} to lowest terms by extracting and canceling out 2.

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