Question

use identities to write the following expression as a function of z alone.
(\tan left(45^{circ}+z
ight))
choose the correct answer below.
a. (\frac{1 + 2 \tan z}{1 - 2 \tan z}) b. (\frac{1 - \tan z}{1 + \tan z})
c. (\frac{1 + \tan z}{1 - \tan z}) d. (\frac{1 - 2 \tan z}{1 + 2 \tan z})

Explanation:

Step1: Use the tangent addition formula

The tangent addition formula is \(\tan(A + B)=\frac{\tan A+\tan B}{1-\tan A\tan B}\). Here \(A = 45^{\circ}\) and \(B=z\), and \(\tan45^{\circ}=1\).
So \(\tan(45^{\circ}+z)=\frac{\tan45^{\circ}+\tan z}{1 - \tan45^{\circ}\tan z}\).

Step2: Substitute \(\tan45^{\circ}=1\)

Substitute \(\tan45^{\circ}=1\) into the formula: \(\frac{1+\tan z}{1-(1)\tan z}=\frac{1 + \tan z}{1-\tan z}\).

Answer:

C. \(\frac{1+\tan z}{1-\tan z}\)