use identities to write the following expression as a function of z alone.\n(\tan left(45^{circ}+z\right))\nc…

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

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

Answer

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})