Question

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

Answer 1

See explanation

Explanation:

Use formula

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

Substitute it into the first fraction:

`(\cos (x+y))/(\cos x\cos y)\\ \\=(\cos x\cos y-\sin x\sin y)/(\cos x\cos y)\\ \\=(\cos x\cos y)/(\cos x\cos y)-(\sin x\sin y)/(\cos x\cos y)\\ \\=1-(\sin x)/(\cos x)\cdot (\sin y)/(\cos y)\\ \\=1-\tan x\tany`

Consider the whole expression:

`(\cos (x+y))/(\cos x\cos y)+\tan x\tan y\\ \\=1-\tan x \tan y+\tan x\tan y\\ \\=1`

Done!