Question

use the sum and difference identities to rewrite the following expression as a trigonometric function of one angle.
\\(\frac{\tan(\frac{3\pi}{4})-\tan(\frac{3\pi}{14})}{1 + \tan(\frac{3\pi}{4})\tan(\frac{3\pi}{14})}\\)

Explanation:

Step1: Recall the tangent - difference identity

The tangent - difference identity is $\tan(A - B)=\frac{\tan A-\tan B}{1 + \tan A\tan B}$.

Step2: Identify $A$ and $B$

In the given expression $\frac{\tan(\frac{3\pi}{4})-\tan(\frac{3\pi}{14})}{1+\tan(\frac{3\pi}{4})\tan(\frac{3\pi}{14})}$, we can identify $A = \frac{3\pi}{4}$ and $B=\frac{3\pi}{14}$.

Step3: Rewrite the expression

Using the tangent - difference identity, the given expression can be rewritten as $\tan(\frac{3\pi}{4}-\frac{3\pi}{14})$.

Answer:

$\tan(\frac{3\pi}{4}-\frac{3\pi}{14})$