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

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Answer 1 · 2016-04-10T02:04:25

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

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!

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