Question

Cos(x+y)/cosxcosy+tanxtany=1

Answer 1

Answer:

See explanation

Step-by-step explanation:

Use formula

$\cos (x+y)=\cos x\cos y-\sin x\sin y$

Substitute it into the first fraction:

$\dfrac{\cos (x+y)}{\cos x\cos y}\\ \\=\dfrac{\cos x\cos y-\sin x\sin y}{\cos x\cos y}\\ \\=\dfrac{\cos x\cos y}{\cos x\cos y}-\dfrac{\sin x\sin y}{\cos x\cos y}\\ \\=1-\dfrac{\sin x}{\cos x}\cdot \dfrac{\sin y}{\cos y}\\ \\=1-\tan x\tany$

Consider the whole expression:

$\dfrac{\cos (x+y)}{\cos x\cos y}+\tan x\tan y\\ \\=1-\tan x \tan y+\tan x\tan y\\ \\=1$

Done!