(3 cos³θ - 2cos θ)/(sinθ - 3sin³θ) is equal to:

Simplify the expression:

(3 cos³θ - 2 cos θ) / (sin θ - 3 sin³θ)

Solution:

Let us use trigonometric identities to simplify the given expression.

We know the following trigonometric identity:

cos³θ = cos θ (1 - sin²θ)

sin³θ = sin θ (1 - cos²θ)

Substitute these identities into the expression:

Numerator:

3 cos³θ - 2 cos θ

= 3 cos θ (1 - sin²θ) - 2 cos θ

= 3 cos θ - 3 cos θ sin²θ - 2 cos θ

= (3 cos θ - 2 cos θ) - 3 cos θ sin²θ

= cos θ - 3 cos θ sin²θ

Denominator:

sin θ - 3 sin³θ

= sin θ - 3 sin θ (1 - cos²θ)

= sin θ - 3 sin θ + 3 sin θ cos²θ

= -2 sin θ + 3 sin θ cos²θ

We notice that:

cos θ = 1 - sin²θ and sin²θ = 1 - cos²θ.

Combine the simplified numerator and denominator:

Numerator: cos θ - 3 cos θ sin²θ

Denominator: -2 sin θ + 3 sin θ cos²θ

By substituting and simplifying, we find:

The expression simplifies to: (cos θ / sin θ) = cot θ

Final Answer: The expression (3 cos³θ - 2 cos θ) / (sin θ - 3 sin³θ) is equal to cot θ.

Shortcut Trick Put θ = 90, then cot2θ = 1/0, undefined and cotθ = 0.

And the expression on putting is also 0.

So, answer is cotθ.

option a and b cannot be answer as given expression is in terms of cosθ / sinθ.