Question

find the exact value of the expression.
(\tan\frac{7pi}{12})
rewrite the expression using a sum or difference formula. choose the correct answer below.
a. (\tan\frac{7pi}{12}=\tan(\frac{pi}{3}+\frac{pi}{4})=\frac{\tan\frac{pi}{3}-\tan\frac{pi}{4}}{1 + \tan\frac{pi}{3}\tan\frac{pi}{4}})
b. (\tan\frac{7pi}{12}=\tan(\frac{pi}{3}-\frac{pi}{4})=\frac{\tan\frac{pi}{3}-\tan\frac{pi}{4}}{1 + \tan\frac{pi}{3}\tan\frac{pi}{4}})
c. (\tan\frac{7pi}{12}=\tan(\frac{pi}{3}+\frac{pi}{4})=\frac{\tan\frac{pi}{3}+\tan\frac{pi}{4}}{1 - \tan\frac{pi}{3}\tan\frac{pi}{4}})
d. (\tan\frac{7pi}{12}=\tan(\frac{pi}{3}-\frac{pi}{4})=\frac{\tan\frac{pi}{3}+\tan\frac{pi}{4}}{1 - \tan\frac{pi}{3}\tan\frac{pi}{4}})

Explanation:

Step1: Recall the sum formula for tangent

The sum formula for tangent is \(\tan(A + B)=\frac{\tan A+\tan B}{1 - \tan A\tan B}\).

Step2: Express \(\frac{7\pi}{12}\) as a sum of known angles

We know that \(\frac{7\pi}{12}=\frac{\pi}{3}+\frac{\pi}{4}\).

Step3: Apply the sum formula

Substitute \(A = \frac{\pi}{3}\) and \(B=\frac{\pi}{4}\) into the sum formula \(\tan(A + B)=\frac{\tan A+\tan B}{1 - \tan A\tan B}\). So \(\tan\frac{7\pi}{12}=\tan(\frac{\pi}{3}+\frac{\pi}{4})=\frac{\tan\frac{\pi}{3}+\tan\frac{\pi}{4}}{1-\tan\frac{\pi}{3}\tan\frac{\pi}{4}}\)

Answer:

C. \(\tan\frac{7\pi}{12}=\tan(\frac{\pi}{3}+\frac{\pi}{4})=\frac{\tan\frac{\pi}{3}+\tan\frac{\pi}{4}}{1 - \tan\frac{\pi}{3}\tan\frac{\pi}{4}}\)