How do you simplify the expression $sec^2theta-tan^2theta+cot^2theta$ ?

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Answer 1 · 2016-09-21T13:28:49

This expression simplifies to $csc^2theta$ .

Apply the following identities:

• $sectheta = 1/costheta$
• $tantheta = sintheta/costheta$
• $cottheta = costheta/sintheta$

$=1/cos^2theta - sin^2theta/cos^2theta + cos^2theta/sin^2theta$

$=(1 - sin^2theta)/(cos^2theta) + cos^2theta/sin^2theta$

Apply the pythagorean identity $sin^2x + cos^2x = 1 -> 1 - sin^2x = cos^2x$

$=cos^2theta/cos^2theta + cos^2theta/sin^2theta$

$= 1 + cos^2theta/sin^2theta$

$=(sin^2theta + cos^2theta)/sin^2theta$

Apply the identity $sin^2beta + cos^2beta = 1$ :

$=1/sin^2theta$

Apply the identity $1/sinalpha = cscalpha$ .

$=csc^2theta$

Hopefully this helps!

How do you simplify the expression sec^2theta-tan^2theta+cot^2thetasec^2theta-tan^2theta+cot^2theta?