Question 1018836: how to prove this,
cosec ^ 4 Theta - cot ^ 4 Theta = 1 + cos ^ 2 Theta/1 - cos ^ 2 Theta
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! this is a difference of squares on the left. I will write x instead of theta and use ctn rather than cot, but that won't affect the result.
csc^4 x-ctn^4 x=
(csc^2 x+ctn^2 x)(csc^2 x- ctn^2 x)=
convert to sin and cos forms, and first parentheses is
(1/sin^2 x)+ (cos^2 x/sin^2 x)=
(cos^2 x+ 1)/sin^2 x, putting it over a common denominator sin^2 x
The other parentheses is
(1/sin^2 x)- (cos^2 x/sin^2 x)
and that is
(1-cos^2 x)/sin^2 x
Their product is (cos^2 x+ 1)/sin^2 x*(1-cos^2 x)/sin^2 x.
But 1-cos^2x=sin^2 x
so the last part of the above is sin^2 x/sin^2 x or 1
The product is (cos^2 x+ 1)/sin^2 x
But sin^2 x= 1-cos^2 x
Therefore, the product above is (1+cos^2 x)/(1-cos^2 x)
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