How do you identify $1/(csctheta+1)-1/(csctheta-1)$ ?

Question

3467 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-04-19T10:10:17

$-2tan^2theta$

I'm going to assume this needs to be simplified:

$1/(csctheta+1)-1/(csctheta-1)$

$1/(1/sintheta+1)-1/(1/sintheta-1)$

$1/(1/sintheta+sintheta/sintheta)-1/(1/sintheta-sintheta/sintheta)$

$1/((1+sintheta)/sintheta)-1/((1-sintheta)/sintheta)$

$sintheta/(1+sintheta)-sintheta/(1-sintheta)$

$sintheta/(1+sintheta)((1-sintheta)/(1-sintheta))-sintheta/(1-sintheta)((1+sintheta)/(1+sintheta))$

$(sintheta(1-sintheta))/((1+sintheta)(1-sintheta))-(sintheta(1+sintheta))/((1-sintheta)(1+sintheta))$

$(sintheta-sin^2theta)/((1+sintheta)(1-sintheta))-(sintheta+sin^2theta)/((1-sintheta)(1+sintheta))$

$(sintheta-sintheta-sin^2theta-sin^2theta)/((1+sintheta)(1-sintheta))$

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

Recall that $sin^2theta+cos^2theta=1=>1-sin^2theta=cos^2theta$

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

How do you identify 1/(csctheta+1)-1/(csctheta-1)1/(csctheta+1)-1/(csctheta-1)?