How do you prove that # cot^2 x +csc^2 x = 2csc^2 x - sin^2x-cos^2x #?

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Answer 1 · 2015-12-20T09:45:03

#csc^2(x) - sin^2(x) - cos^2(x) = 1/sin^2(x) - sin^2(x)-cos^2(x)#

#= 1/sin^2(x) - (sin^2(x) + cos^2(x))#

#= 1/sin^2(x) - 1#

#= 1/sin^2(x) - sin^2(x)/sin^2(x)#

#= (1-sin^2(x))/sin^2(x)#

#= cos^2(x)/sin^2(x)#

#= cot^2(x)#

Then, as

#cot^2(x)=csc^2(x) - sin^2(x) - cos^2(x)#

Adding #csc^2(x)# to both sides gives us

#cot^2(x) + csc^2(x) = 2csc^2(x) - sin^2(x) - cos^2(x)#