Question

Can you please figure out this equation and see if I have the right answer

Answer 1

The expression is given to be:

`\tan \left((7\pi )/(12)\right)`

Rewrite the expression:

`(7\pi)/(12)=(\pi)/(4)+(\pi)/(3)`

Therefore, we have:

`\tan\left((7\pi)/(12)\right)=\tan\left((\pi)/(4)+(\pi)/(3)\right)`

Recall the summation identity:

`\tan \left(x+y\right)=(\tan \left(x\right)+\tan \left(y\right))/(1-\tan \left(x\right)\tan \left(y\right))`

Therefore, we have:

`\tan\left((\pi)/(4)+(\pi)/(3)\right)=(\tan\left((\pi)/(4)\right)+\tan\left((\pi)/(3)\right))/(1-\tan\left((\pi)/(4)\right)\tan\left((\pi)/(3)\right))`

Recall that:

`\begin{gathered} \tan \left((\pi )/(4)\right)=1 \\ \tan \left((\pi )/(3)\right)=√(3) \end{gathered}`

Hence, the equation becomes:

`(\tan((\pi)/(4))+\tan((\pi)/(3)))/(1-\tan(\pi\/4)\tan(\pi\/3))=(1+√(3))/(1-1\cdot√(3))`

Therefore, we can simplify the expression to be:

`-2-√(3)`

The THIRD OPTION is correct.