I need help with number 5

Step-by-step explanation:

We have the following relationships;

We can write;

We note that the numerator of , which is is the derivative of the denominator, , therefore, we can use integration by substitution method and write;

, from which we get;

Therefore, we can write;

Therefore;

csc x + cot x = (1/sin x) + ((cos x)/(sin x)) = (1 + cos x)/(sin x)

(1 + cos x)/(sin x) = (cos²(x/2) + sin²(x/2) + cos²(x/2) - sin²(x/2))/(2sin(x/2)·cos(x/2)) = (2·cos²(x/2))/((2sin(x/2)·cos(x/2)) =  cos(x/2)/sin(x/2) = cot(x/2)

Therefore;

Therefore;