Question

Rewrite the given expression as the product of powers of sinθ and cosθ using negative exponents for terms in the denominator.

[csc ⁴ (θ)]/ cot ⁵ (θ)

Answer 1

To rewrite csc⁴(θ)/cot⁵(θ) using powers of sinθ and cosθ with negative exponents, transform cscθ to 1/sinθ and cotθ to cosθ/sinθ, then apply negative exponent rules to get sinθ/cosθ⁵.

Step-by-step explanation:

To rewrite the expression csc ⁴ (θ) / cot ⁵ (θ) using the powers of sinθ and cosθ with negative exponents, we need to express the cosecant and cotangent in terms of sine and cosine and then apply the rules of negative exponents and division of exponentials.

The cosecant function is the reciprocal of the sine function, so csc θ = 1/sin θ. Similarly, the cotangent function is the reciprocal of the tangent function, which is cosine over sine, so cot θ = cos θ/sin θ. Applying these transformations gives:

csc ⁴ (θ) / cot ⁵ (θ) = (1/sinθ) ⁴ / (cosθ/sinθ) ⁵

Transforming to negative exponents, we have:

(sinθ)-4 / {(cosθ)5 / (sinθ)5}

By dividing the exponential terms, we get:

(sinθ)-4 * (sinθ)5 / (cosθ)5

Finally, we subtract the exponents when multiplying powers of the same base:

sinθ1 / cosθ5