Question

Differentiate sinθ / (1 - cosθ).

a) (1 + cosθ) / (sinθ)
b) tanθ / (1 - cosθ)
c) cotθ / (sinθ)
d) cscθ / (cosθ)

Answer 1

The derivative of sinθ / (1 - cosθ) can be found using the quotient rule and simplifying with the Pythagorean identity. The final derivative simplifies to (1 + cosθ) / (sinθ), which matches option (a).

Step-by-step explanation:

The student has asked to differentiate the function sinθ / (1 - cosθ). To find the derivative, we will use the quotient rule which states that the derivative of a function f(x)/g(x) is (f'(x)g(x) - f(x)g'(x))/(g(x))^2.

Let's set u = sinθ and v = 1 - cosθ. Then u' = cosθ and v' = sinθ since the derivative of sinθ is cosθ and the derivative of -cosθ is sinθ.

Applying the quotient rule, we get:

∂(u/v) = (u'v - uv')/v^2

∂(sinθ/(1-cosθ)) = (cosθ(1-cosθ) - sinθ(-sinθ)) / (1-cosθ)^2

∂(sinθ/(1-cosθ)) = (cosθ - cos^2θ + sin^2θ) / (1 - 2cosθ + cos^2θ)

Using the Pythagorean identity, sin^2θ + cos^2θ = 1, we can simplify the numerator to:

(1 - cosθ + sin^2θ) / (1 - 2cosθ + cos^2θ)

This simplifies further to:

(1 + sin^2θ) / (1 - 2cosθ + cos^2θ) = (1 + sin^2θ) / (sin^2θ)

Finally, the derivative is:

∂(sinθ/(1-cosθ)) = (1 + sin^2θ) / sin^2θ

Option (a) (1 + cosθ) / (sinθ) is the correct answer upon simplification.