How do you prove $csc^2theta-cot^2theta=cotthetatantheta$ ?

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Answer 1 · 2016-12-27T04:21:00

Use the identities $cscbeta = 1/sinbeta$ , $cotbeta = cosbeta/sinbeta$ and $tanbeta = sin beta/cosbeta$ .

$1/sin^2theta - cos^2theta/sin^2theta = costheta/sintheta * sintheta/costheta$

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

Use the identity $sin^2beta + cos^2beta = 1$ .

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

$1 = 1$

$LHS = RHS$

Hopefully this helps!

Answer 2 · 2016-12-27T04:35:07

$csc^2theta-cot^2theta=cotthetatantheta$

Use the Pythagorean identity $1+cot^2theta=csc^2theta$

$1+cot^2theta -cot^2theta=cotthetatantheta$

$1=cotthetatantheta$

$tantheta/tantheta=cotthetatantheta$

$tantheta * 1/tantheta =cotthetatantheta$

Use the identity $cottheta=1/tantheta$

$tantheta * cottheta= cotthetatantheta$

$cotthetatantheta=cotthetatantheta$

How do you prove csc^2theta-cot^2theta=cotthetatanthetacsc^2theta-cot^2theta=cotthetatantheta?