What is the integral of ([cos(θ-1)])/([sin^2(θ-1)]) ?

The integral of ([cos(θ-1)])/([sin^2(θ-1)]) is -csc(θ-1)+C

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integral of ([cos(θ-1)])/([sin^2(θ-1)])

\int \frac{[ cos ( θ - 1 ) ]}{[ sin ^{2} ( θ - 1 ) ]} d θ

- csc (θ - 1) + C

Solution steps

\int \frac{[ cos ( θ - 1 ) ]}{[ sin ^{2} ( θ - 1 ) ]} d θ

Rewrite using trig identities

= \int \frac{cos ( θ - 1 )}{( - sin ( 1 ) cos ( θ ) + cos ( 1 ) sin ( θ ) ) ^{2}} d θ

Use the Angle Difference identity:

sin (s) cos (t) - cos (s) sin (t) = sin (s - t)

= \int \frac{cos ( θ - 1 )}{sin ^{2} ( θ - 1 )} d θ

Apply u-substitution

: \int \frac{cot ( u )}{sin ( u )} d u

= \int \frac{cot ( u )}{sin ( u )} d u

Rewrite using trig identities

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

Apply u-substitution

: \int \frac{1}{v ^{2}} d v

= \int \frac{1}{v ^{2}} d v

Apply the Power Rule

: - \frac{1}{v}

= - \frac{1}{v}

Substitute back

= - \frac{1}{sin ( θ - 1 )}

Use the basic trigonometric identity:

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

= - csc (θ - 1)

Add a constant to the solution

= - csc (θ - 1) + C

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What is the integral of ([cos(θ-1)])/([sin^2(θ-1)]) ?
The integral of ([cos(θ-1)])/([sin^2(θ-1)]) is -csc(θ-1)+C