Question

Write the simplest form of tan inverse bracket 3 cos x - 4 sin x by 4 cos x + 3 sin x if 3 by 4 then x is greater than minus 1

Answer 1

Answer:

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Answer 2

Explanation:

$\tan^-^1\left ( \frac{3\cos x-4 \sin x}{4 \cos x+3 \sin x} \right )\\\\y = \frac{3\cos x-4 \sin x}{4 \cos x+3 \sin x}$

divide numerator and denominator by 4 cos x

$y = \frac{\frac{3}{4}-\tan x }{1+\frac{3}{4}\tan x }$

let

$\frac{3}{4}=\tan \theta \\\\then \\\\y = \frac{\tan\theta-\tan x }{1+\tan\theta\tan x}\\\\y = \tan (\theta -x )\\\\\tan^-^1 y = \tan^-^1 (\tan (\theta -x ))=(\theta -x )= \tan^-^1(\frac{3}{4})-x$

thus , the simplest form is

$\tan^-^1(\frac{3}{4})-x$

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