Solve 1+tan^230/1-tan^230

$$\frac { 1 + \tan ^ { 2 } 30 } { 1 - \tan ^ { 2 } 30 }$$

Evaluate

$$2$$

Solution Steps

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

$$\frac{1+\left(\frac{\sqrt{3}}{3}\right)^{2}}{1-\left(\tan(30)\right)^{2}}$$

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

$$\frac{1+\frac{\left(\sqrt{3}\right)^{2}}{3^{2}}}{1-\left(\tan(30)\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{3^{2}}{3^{2}}.

$$\frac{\frac{3^{2}}{3^{2}}+\frac{\left(\sqrt{3}\right)^{2}}{3^{2}}}{1-\left(\tan(30)\right)^{2}}$$

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

$$\frac{\frac{3^{2}+\left(\sqrt{3}\right)^{2}}{3^{2}}}{1-\left(\tan(30)\right)^{2}}$$

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

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

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

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

The square of \sqrt{3} is 3.

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

Calculate 3 to the power of 2 and get 9.

$$\frac{\frac{3^{2}+\left(\sqrt{3}\right)^{2}}{3^{2}}}{1-\frac{3}{9}}$$

Reduce the fraction \frac{3}{9} to lowest terms by extracting and canceling out 3.

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

Subtract \frac{1}{3} from 1 to get \frac{2}{3}.

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

Divide \frac{3^{2}+\left(\sqrt{3}\right)^{2}}{3^{2}} by \frac{2}{3} by multiplying \frac{3^{2}+\left(\sqrt{3}\right)^{2}}{3^{2}} by the reciprocal of \frac{2}{3}.

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

Cancel out 3 in both numerator and denominator.

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

The square of \sqrt{3} is 3.

$$\frac{3+3^{2}}{2\times 3}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{3+9}{2\times 3}$$

Add 3 and 9 to get 12.

$$\frac{12}{2\times 3}$$

Multiply 2 and 3 to get 6.

$$\frac{12}{6}$$

Divide 12 by 6 to get 2.

$$2$$