What is the integral of (sin(x))/((sin(cos(x)))^2) ?

The integral of (sin(x))/((sin(cos(x)))^2) is cot(cos(x))+C

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integral of (sin(x))/((sin(cos(x)))^2)

\int \frac{sin ( x )}{( sin ( cos ( x ) ) ) ^{2}} d x

cot (cos (x)) + C

Solution steps

\int \frac{sin ( x )}{( sin ( cos ( x ) ) ) ^{2}} d x

Use the basic trigonometric identity:

\frac{1}{sin ( x )} = csc (x)

= \int csc^{2} (cos (x)) sin (x) d x

Apply u-substitution

: \int - csc^{2} (u) d u

= \int - csc^{2} (u) d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int csc^{2} (u) d u

Use the common integral:

\int csc^{2} (u) d u = - cot (u)

= - (- cot (u))

Substitute back

u = cos (x)

= - (- cot (cos (x)))

Simplify

= cot (cos (x))

Add a constant to the solution

= cot (cos (x)) + C

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What is the integral of (sin(x))/((sin(cos(x)))^2) ?
The integral of (sin(x))/((sin(cos(x)))^2) is cot(cos(x))+C