Answer:
[tex] { \cos}^{3} x[/tex]
Step-by-step explanation:
We want to simplify:
[tex] \frac{ \sin( \frac{\pi}{2} - x) }{ { \cot}^{2} ( \frac{\pi}{2} -1 ) + 1} [/tex]
Use the Pythagorean identity:
[tex] { \csc}^{2}x = { \cot}^{2}x + 1[/tex]
We apply this property to get:
[tex] \frac{ \sin( \frac{\pi}{2} - x) }{ { \csc}^{2} ( \frac{\pi}{2} -x) } [/tex]
This gives us:
[tex]\frac{ \sin( \frac{\pi}{2} - x) }{ \frac{1}{{ \sin}^{2} ( \frac{\pi}{2} -x)} } [/tex]
We simplify to get:
[tex]\sin^{3} ( \frac{\pi}{2} -x)[/tex]
[tex](\sin ( \frac{\pi}{2} -x))^{3}[/tex]
Apply the complementary identity;
[tex](\cos x)^{3} = { \cos}^{3} x[/tex]
