Use a half-angle formula to find the exact value of tan(3pi )/(8) tan(3pi )/(8)=

To find the exact value of $\tan\frac{3\pi}{8}$ using a half-angle formula, we can use the identity:<br /><br />$\tan\left(\frac{x}{2}\right) = \frac{2\tan(x)}{1 - \tan^2(x)}$<br /><br />Let $x = \frac{3\pi}{4}$. Then $\frac{x}{2} = \frac{3\pi}{8}$.<br /><br />We need to find $\tan\left(\frac{3\pi}{8}\right)$, which is $\tan\left(\frac{x}{2}\right)$.<br /><br />First, we need to find $\tan(x)$, where $x = \frac{3\pi}{4}$.<br /><br />Since $\frac{3\pi}{4}$ is in the second quadrant, $\tan\left(\frac{3\pi}{4}\right)$ will be negative.<br /><br />We know that $\tan\left(\frac{3\pi}{4}\right)$ is the negative reciprocal of $\tan\left(\frac{\pi}{4}\right)$.<br /><br />Since $\tan\left(\frac{\pi}{4}\right) = 1$, we have:<br /><br />$\tan\left(\frac{3\pi}{4}\right) = -1$<br /><br />Now, we can use the half-angle formula:<br /><br />$\tan\left(\frac{x}{2}\right) = \frac{2\tan(x)}{1 - \tan^2(x)}$<br /><br />Substituting the values, we get:<br /><br />$\tan\left(\frac{3\pi}{8}\right) = \frac{2(-1)}{1 - (-1)^2} = \frac{-2}{1 - 1} = \frac{-2}{0}$<br /><br />Since division by zero is undefined, we need to re-evaluate our approach.<br /><br />Instead, let's use the half-angle formula directly for $\frac{3\pi}{8}$:<br /><br />$\tan\left(\frac{3\pi}{8}\right) = \sqrt{\frac{1 - \cos\left(\frac{3\pi}{4}\right)}{1 + \cos\left(\frac{3\pi}{4}\right)}}$<br /><br />Since $\cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}$, we have:<br /><br />$\tan\left(\frac{3\pi}{8}\right) = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{1 - \frac{\sqrt{2}}{2}}}$<br /><br />Simplifying the expression inside the square root, we get:<br /><br />\left(\frac{3\pi}{8}\right) = \sqrt{\frac{2 + \sqrt{2}}{2 - \sqrt{2}}}$<br /><br />Rationalizing the denominator, we get:<br /><br />$\tan\left(\frac{3\pi}{8}\right) = \sqrt{\frac{(2 + \sqrt{2})(2 + \sqrt{2})}{(2 - \sqrt{2})(2 + \sqrt{2})}}$<br /><br />Simplifying further, we get:<br /><br />$\tan\left(\frac{3\pi}{8}\right) = \sqrt{\frac{4 + 2\sqrt{2} + 2\sqrt{2} + 2}{4 - 2}}$<br /><br />Combining like terms, we get:<br /><br />$\tan\left(\frac{3\pi}{8}\right) = \sqrt{\frac{6 + 4\sqrt{2}}{2}}$<br /><br />Simplifying the square root, we get:<br /><br />$\tan\left(\frac{3\pi}{8}\right) = \sqrt{3 + 2\sqrt{2}}$<br /><br />Therefore, the exact value of $\tan\left(\frac{3\pi}{8}\right)$ is $\sqrt{3 + 2\sqrt{2}}$.