How to verify the trigonometric identity (tan θ) + (cot θ) / (sec^2 θ) = cot θ

LHS = (tan θ) + (cot θ) / (sec²θ)

First simplify the numerator:

(tan θ) + (cot θ) = tanθ + (1/tanθ) = (1+tan²θ)/tanθ

Since 1+tan²θ = sec²θ, you can rewrite (1+tan²θ)/tanθ as sec²θ/tanθ

So on the left hand side you have

LHS = (sec²θ/tanθ)/sec²θ = (sec²θ/tanθ)*(1/sec²θ)

By cancelling out sec²θ you get

LHS = 1/tanθ = cotθ

first less look at cotx/sec^2x. cotx=cosx/sinx and sec^2x=1/cos^2x

cosx/sinx*cos^2x=cos^3x/sinx

tanx+(cos^3x)/(sinx)
tanx=sinx/cosx
now multiply sinx/cosx by sinx/sinx and multiply cos^3x/sinx by cosx/cosx

[sin^2x+cos^4x]/(sinxcosx)
[1-cos^2x+cos^4x]/(sinxcosx)

from tanx + cotx = 1/(sinx cosx)
and sec^2 x = 1/cos^2 x

left hand side = cosx/sinx = cot x <---right hand side

wat