Solve 2tan^{2}45^{circ}+cos^230^circ-sin^260^circ

$$2 \tan ^ { 2 } 45 ^ { \circ } + \cos ^ { 2 } 30 ^ { \circ } - \sin ^ { 2 } 60 ^ { \circ }$$

Evaluate

$$2$$

Solution Steps

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

$$2\times 1^{2}+\left(\cos(30)\right)^{2}-\left(\sin(60)\right)^{2}$$

Calculate 1 to the power of 2 and get 1.

$$2\times 1+\left(\cos(30)\right)^{2}-\left(\sin(60)\right)^{2}$$

Multiply 2 and 1 to get 2.

$$2+\left(\cos(30)\right)^{2}-\left(\sin(60)\right)^{2}$$

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

$$2+\left(\frac{\sqrt{3}}{2}\right)^{2}-\left(\sin(60)\right)^{2}$$

To raise \frac{\sqrt{3}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$2+\frac{\left(\sqrt{3}\right)^{2}}{2^{2}}-\left(\sin(60)\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{2^{2}}{2^{2}}.

$$\frac{2\times 2^{2}}{2^{2}}+\frac{\left(\sqrt{3}\right)^{2}}{2^{2}}-\left(\sin(60)\right)^{2}$$

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

$$\frac{2\times 2^{2}+\left(\sqrt{3}\right)^{2}}{2^{2}}-\left(\sin(60)\right)^{2}$$

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

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

To raise \frac{\sqrt{3}}{2} to a power, raise both numerator and denominator to the power and then divide.

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

The square of \sqrt{3} is 3.

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

Calculate 2 to the power of 2 and get 4.

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

To add or subtract expressions, expand them to make their denominators the same. Expand 2^{2}.

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

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

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

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

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

Calculate 2 to the power of 3 and get 8.

$$\frac{8+\left(\sqrt{3}\right)^{2}}{2^{2}}-\frac{3}{4}$$

The square of \sqrt{3} is 3.

$$\frac{8+3}{2^{2}}-\frac{3}{4}$$

Add 8 and 3 to get 11.

$$\frac{11}{2^{2}}-\frac{3}{4}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{11}{4}-\frac{3}{4}$$

Subtract \frac{3}{4} from \frac{11}{4} to get 2.

$$2$$