Question

Use the sum and difference identities to rewrite the following expression as a trigonometric function of a single number.tan(5014+ tantan (14)tan(1)571 - tan

Answer 1

From the sum identity of the tangent function given by

`\tan (\alpha+\beta)=(\tan\alpha+\tan\beta)/(1-\tan\alpha\cdot\tan\beta)`

and by comparing this fomula with our expression, we can note that

`\begin{gathered} \alpha=(5\pi)/(14) \\ \text{and} \\ \beta=(\pi)/(14) \end{gathered}`

Then by substituting these values into the formula, we get

`\tan ((5\pi)/(14)+(\pi)/(14))=(\tan(5\pi)/(14)+\tan(\pi)/(14))/(1-\tan(5\pi)/(14)\cdot\tan(\pi)/(14))`

Since

`(5\pi)/(14)+(\pi)/(14)=(6\pi)/(14)=(3\pi)/(7)`

the answer is:

`tan(3\pi)/(7)`