How can this be simplified to the most basic form?

Question

#tan^2xcsc^2x - tan^2x#

Please include detailed explanation of the process, Thank you!

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Answer 1 · 2018-03-02T15:59:32

#1#

Starting with what we have

#=>tan^2 x *cosec^2 x -tan^2 x#

#=>(cancel (sin^2x))/(cos^2x)*1/cancel(sin^2x)-tan^2 x#

#=>sec^2x-tan^2x=1#

#color (green)(sec^2x-tan^2x#

#=>color (green)(1/(cos^2x)-(sin^2x)/(cos^2x))#

#=>color (green)((1-sin^2x)/(cos^2) #

Using the identity

#color (blue)(cos^2x + sin^2x = 1)#

#color (blue)(cos^2x = 1-sin^2x)#

We get

#color (green)((cos^2x)/(cos^2x))#

#=>color (red)1#