Question #b725f

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Answer 1 · 2017-04-01T21:52:33

I assume you're asking to verify the identity:

#tan^2theta/(sectheta-1)^2=(1+costheta)/(1-costheta)#

Modify the left-hand side only. Use the identity #tan^2theta+1=sec^2theta#, so #tan^2theta=sec^2theta-1#:

#tan^2theta/(sectheta-1)^2=(sec^2theta-1)/(sectheta-1)^2#

We can factor #sec^2theta-1# as a difference of squares:

#(sec^2theta-1)/(sectheta-1)^2=((sectheta+1)(sectheta-1))/(sectheta-1)^2=(sectheta+1)/(sectheta-1)#

Recall that #sectheta=1/costheta#:

#(sectheta+1)/(sectheta-1)=(1/costheta+1)/(1/costheta-1)#

Simplify this by multiplying it by #costheta/costheta#:

#(1/costheta+1)/(1/costheta-1)=(costheta(1/costheta+1))/(costheta(1/costheta-1))=(1+costheta)/(1-costheta)#

Since this is the right-hand side, we've proven this identity.