Question

verify the identity.
\\( \tan ( 2 \pi - \theta ) = - \tan \theta \\)
write the left side of the identity using either the sum or difference formula for tangent.

Explanation:

Step1: Use the tangent difference formula

The tangent difference formula is $\tan(A - B)=\frac{\tan A-\tan B}{1 + \tan A\tan B}$. Let $A = 2\pi$ and $B=\theta$. Then $\tan(2\pi-\theta)=\frac{\tan2\pi-\tan\theta}{1+\tan2\pi\tan\theta}$.

Step2: Simplify using the value of $\tan2\pi$

Since $\tan2\pi = 0$, we substitute it into the formula: $\frac{0-\tan\theta}{1+0\times\tan\theta}$.

Step3: Calculate the result

Simplify the expression $\frac{0-\tan\theta}{1+0\times\tan\theta}=\frac{-\tan\theta}{1}=-\tan\theta$.

Answer:

The left - hand side $\tan(2\pi-\theta)$ simplifies to $-\tan\theta$, which is equal to the right - hand side of the given identity. So the identity $\tan(2\pi-\theta)=-\tan\theta$ is verified.