Chapter 7.3, Problem 89E

To determine

To prove the identity:

  $\frac{\sin x+\sin y}{\cos x+\cos y}=\tan \frac{x+y}{2}$

Explanation of Solution

Given:

An identity is given as

  $\frac{\sin x+\sin y}{\cos x+\cos y}=\tan \frac{x+y}{2}$

Concept Used:

Consider the sum-to-product formula.

  $\sin u+\sin v=2\sin \left(\frac{u+v}{2}\right)\cos \left(\frac{u-v}{2}\right)$

Consider the sum-to-product formula.

  $\cos u+\cos v=2\cos \left(\frac{u+v}{2}\right)\cos \left(\frac{u-v}{2}\right)$

Quotient Identity

  $\tan x=\frac{\sin x}{\cos x}$

Calculation:

In order to verify the identity:

  $\frac{\sin x+\sin y}{\cos x+\cos y}=\tan \frac{x+y}{2}$

Take left hand of above identity and simplify it using the trigonometric identities.

As,

  $\begin{array}{c}\frac{\sin x+\sin y}{\cos x+\cos y}=\frac{2\sin \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)}{2\cos \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)}\\ =\tan \frac{x+y}{2}\\ =\text{Right hand}\end{array}$

Thus, identity is verified, i.e.

  $\frac{\sin x+\sin y}{\cos x+\cos y}=\tan \frac{x+y}{2}$