Question

Perform the addition and simplify: tanx/cscx + sinx/tanx

Answer 1

`(\tan x)/(\csc x)+(\sin x)/(\tan x)=(\sin x)/(\cos x):(1)/(\sin x)+\sin x:(\sin x)/(\cos x)\\\\=(\sin x)/(\cos x)\cdot(\sin x)/(1)+\sin x\cdot(\cos x)/(\sin x)=(\sin^2x)/(\cos x)+\cos x\\\\=(\sin^2x)/(\cos x)+(\cos x\cdot\cos x)/(\cos x)=(\sin^2x)/(\cos x)+(\cos^2x)/(\cos x)\\\\=(\sin^2x+\cos^2x)/(\cos x)=(1)/(\cos x)=\sec x\\\\Answer:\boxed{(\tan x)/(\csc x)+(\sin x)/(\tan x)=\sec x}`



`Used:\\\csc x=(1)/(\sin x);\ \sec x=(1)/(\cos x)\\\\\tan x=(\sin x)/(\cos x)\\\\\sin^2x+\cos^2x=1`

Answer 2

[tan/csc] + [sin/tan)]
=[(sin/cos)/(1/sin)] + [sin/(sin/cos)]
=[sin^2/cos] + [cos]
=[sin^2/cos] + [cos^2/cos]
=[(sin^2+cos^2]/cos
=1/cos
=sec