How do you verify #(csc x+sec x)/(sin x+ cos x)=cot x+tan x#?

1 Answer
Apr 10, 2016

For an identity like this, we have to be clear with the following identities.

Explanation:

The reciprocal identities

#csctheta = 1/sintheta#

#sectheta = 1/costheta#

#cottheta = 1/tantheta#

The quotient identities:

#tantheta = sin theta/costheta#

#cottheta = costheta/sintheta#

Applying all these identities, on both sides, we get:

#(1/sinx + 1/cosx)/(sinx + cosx) = cosx/sinx + sinx/cosx#

#((cosx + sinx)/(cosxsinx))/(sinx + cosx) = cosx/sinx + sinx/cosx#

#1/(sinx + cosx) xx (cosx + sinx)/(cosxsinx) = cosx/sinx + sinx/cosx#

#1/(cosxsinx)= (cos^2x + sin^2x)/(sinxcosx)#

Applying the pythagorean identity #sin^2x + cos^2x = 1# on the right side, we get:

#1/(cosxsinx) = 1/(sinxcosx)#

Hopefully this helps!