Chapter 7, Problem 44RE

In Problems 28-44, establish each identity.

$\frac{\cos \left(2\theta \right)-\cos \left(4\theta \right)}{\cos \left(2\theta \right)+\cos \left(4\theta \right)}-\tan \theta \tan \left(3\theta \right)=0$

To determine

To prove: Establish each identity:

Answer to Problem 44RE

$\frac{\text{cos}2\theta -\text{ cos}4\theta }{\text{cos}2\theta +\text{ cos}4\theta }-\text{ tan}\theta \text{tan}3\theta =0$

Explanation of Solution

Given:

$\frac{\text{cos}2\theta -\text{ cos}4\theta }{\text{cos}2\theta +\text{ cos}4\theta }-\text{ tan}\theta \text{tan}3\theta =0$

Proof:

$\text{L}\text{.H}\text{.S = }\frac{\text{cos}2\theta -\text{ cos}4\theta }{\text{cos}2\theta +\text{ cos}4\theta }-\text{ tan}\theta \text{tan}3\theta$

$\frac{\text{cos}2\theta -\text{ cos}4\theta }{\text{cos}2\theta +\text{ cos}4\theta }-\text{ tan}\theta \text{tan}3\theta =\frac{-2\text{sin}3\theta \text{cos}\left(-\theta \right)}{2\text{cos}3\theta \text{cos}\left(-\theta \right)}-\text{ tan}\theta \text{tan}3\theta =0$

$\text{L}\text{.H}\text{.S = R}\text{.H}\text{.S}$

Hence Proved.