Prove the identity (sin^3(x) + cos^3(x))/(1 - 2 cos²(x)) = (sec(x) - sin(x))/(tan(x) - 1)

Refresh if you see display errors such as \begin{aligned}, etc.

is that symbolab i see in the picture?

left-hand side:

   

if we factor out a negative from the denominator

   

we have a cosine double angle identity, which states that  . converting that denominator, we get

   

for the numerator of  , we can factor. this is the sum of cubes factoring, which is 

   

with  and  we factor the numerator

   

since  , we have

   



right-hand side:

   

one strategy is to change everything in terms of sine and cosine.
since we have  and  we get

   

we can get rid of the little cosines in the nested fractions by multiplying numerator and denominator the bigger fraction by cos x, which will distribute and cancel out the cosines accordingly. it's the same as multiplying by 1, which doesn't change the expression.

   

if we take a look back at the left-side for a moment here

   

another form of cosine double angle identity:   .

so we have to do something to the right-hand side's  to get  in the denominator to use the cosine double angle identity. then we will be on the right track (get it, right? haha rightside)

another strategy we can use is to multiply everything by the conjugate.
the conjugate of the denominator of  is 

multiplying an expression by its conjugate produces a difference of squares, i.e.

   

so it follows that

   

creating a difference squares will let us use the cosine double angle identity; we just need to factor out a negative.

(another rationale you could've made: seeing that the left-hand side has  while the right-hand side doesn't, you could try to force  on to the right-hand side by multiplying the numerator and denominator by it to see what happens.)

let us take the right-hand side

   

and multiply the numerator and denominator by the conjugate of the denominator. (we have to multiply it to the  numerator and denominator to not change the original expression.)

   

if we factor a negative from the denominator, we can use the cosine double angle identity: