We have: #frac(tan^(2)(x) csc^(2)(x) - 1)(csc(x) tan^(2)(x) sin(x))#
Let's apply the standard trigonometric identities #tan(x) = frac(sin(x))(cos(x))# and #csc(x) = frac(1)(sin(x)#:
#= frac(frac(sin^(2)(x))(cos^(2)(x)) cdot frac(1)(sin^(2)(x)) - 1)(csc(x) tan^(2)(x) sin(x))#
#= frac(frac(1)(cos^(2)(x)) - 1)(csc(x) tan^(2)(x) sin(x))#
Then, let's apply another standard trigonometric identity; #sec(x) = frac(1)(cos(x))#:
#= frac(sec^(2)(x) - 1)(csc(x) tan^(2)(x) sin(x))#
One of the Pythagorean identities is #tan^(2)(x) + 1 = sec^(2)(x)#.
We can rearrange it to get:
#Rightarrow tan^(2)(x) = sec^(2)(x) - 1#
Let's apply this rearranged identity to our proof:
#= frac(tan^(2)(x))(csc(x) tan^(2)(x) sin(x))#
#= frac(1)(csc(x) sin(x))#
Finally, let's apply the standard trigonometric identity #csc(x) = frac(1)(sin(x))#:
#= frac(1)(frac(1)(sin(x)) cdot sin(x))#
#= frac(1)(1)#
#= 1#
#therefore frac(tan^(2)(x) csc^(2)(x) - 1)(csc(x) tan^(2)(x) sin(x)) = 1#
