Solve 3tanpi/8/1-tan^2frac{pi{8}}=

$$\frac { 3 \tan \frac { \pi } { 8 } } { 1 - \tan ^ { 2 } \frac { \pi } { 8 } } =$$

Evaluate

$$\frac{3}{2}=1.5$$

Solution Steps

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

$$\frac{3\left(\sqrt{2}-1\right)}{1-\left(\tan(\frac{\pi }{8})\right)^{2}}$$

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{2}-1\right)^{2}.

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

The square of \sqrt{2} is 2.

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

Add 2 and 1 to get 3.

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

To find the opposite of 3-2\sqrt{2}, find the opposite of each term.

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

Subtract 3 from 1 to get -2.

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

Rationalize the denominator of \frac{3\left(\sqrt{2}-1\right)}{-2+2\sqrt{2}} by multiplying numerator and denominator by -2-2\sqrt{2}.

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

Consider \left(-2+2\sqrt{2}\right)\left(-2-2\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Calculate -2 to the power of 2 and get 4.

$$\frac{3\left(\sqrt{2}-1\right)\left(-2-2\sqrt{2}\right)}{4-\left(2\sqrt{2}\right)^{2}}$$

Expand \left(2\sqrt{2}\right)^{2}.

$$\frac{3\left(\sqrt{2}-1\right)\left(-2-2\sqrt{2}\right)}{4-2^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{3\left(\sqrt{2}-1\right)\left(-2-2\sqrt{2}\right)}{4-4\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{3\left(\sqrt{2}-1\right)\left(-2-2\sqrt{2}\right)}{4-4\times 2}$$

Multiply 4 and 2 to get 8.

$$\frac{3\left(\sqrt{2}-1\right)\left(-2-2\sqrt{2}\right)}{4-8}$$

Subtract 8 from 4 to get -4.

$$\frac{3\left(\sqrt{2}-1\right)\left(-2-2\sqrt{2}\right)}{-4}$$

Use the distributive property to multiply 3 by \sqrt{2}-1.

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

Use the distributive property to multiply 3\sqrt{2}-3 by -2-2\sqrt{2} and combine like terms.

$$\frac{-6\left(\sqrt{2}\right)^{2}+6}{-4}$$

The square of \sqrt{2} is 2.

$$\frac{-6\times 2+6}{-4}$$

Multiply -6 and 2 to get -12.

$$\frac{-12+6}{-4}$$

Add -12 and 6 to get -6.

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

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

$$\frac{3}{2}$$

Factor

$$\frac{3}{2} = 1\frac{1}{2} = 1.5$$