SOLUTION: how to prove this, cosec ^ 4 Theta - cot ^ 4 Theta = 1 + cos ^ 2 Theta/1

SOLUTION: how to prove this, cosec ^ 4 Theta - cot ^ 4 Theta = 1 + cos ^ 2 Theta/1 - cos ^ 2 Theta

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)
RELATED QUESTIONS
Verify Cos(theta) / (1+cos(theta)) + cos(theta)/(1-cos(theta)) = 2 cot(theta)... (answered by Edwin McCravy)

prove that tan theta +4 / 4 cot theta +1 = sin theta/cos theta (answered by stanbon)

establish each identity A) sec^4 theta - sec^2 theta = tan^4 theta + tan^2 theta (answered by solver91311)

Simplify each trig expression into 1 term: 1.) sin(theta)+cos(theta)cot(theta)... (answered by stanbon)

prove that sin^6 theta = -1/32 (cos 6 theta - 6 cos 4 theta + 16 cos 2 theta -... (answered by Edwin McCravy)

prove that cos 7 theta +cos 5 theta + cos 3 theta + cos theta = 4 cos theta cos 2 theta... (answered by Edwin McCravy)

Please help me prove these following identities with explanation, thanks <3 1.) {{{... (answered by Alan3354)

Prove: cos^2 theta/1-sin theta=1+sin... (answered by stanbon,lwsshak3)

Prove the following identity : (cos^2 theta - sin^2 theta)/(cos^2 theta + sin theta cos... (answered by MathLover1)