Final answer:
The trigonometric expression cos θ/(1-csc θ) multiplied by (1+csc θ)/(1+csc θ) simplifies to -cos θ, after applying trigonometric identities and the algebraic technique of multiplying by a conjugate.
Step-by-step explanation:
You want to simplify the trigonometric expression cos θ / (1 - csc θ) by multiplying it with (1 + csc θ) / (1 + csc θ). This is a standard algebraic technique often used in trigonometry to simplify expressions by multiplying the numerator and denominator by a conjugate. Since csc θ is the reciprocal of sin θ, meaning that csc θ = 1/sin θ, we can rewrite the expression using sine.
First, you should rewrite csc θ in terms of sine:
csc θ = 1/sin θ
Then the original expression becomes:
cos θ / (1 - 1/sin θ) which is cos θ / (sin θ - 1) / sin θ
Now multiply the top and bottom by (1 + csc θ) which is (1 + 1/sin θ):
(cos θ * (1 + 1/sin θ)) / ((sin θ - 1) * (1 + 1/sin θ)) / sin θ
This expression simplifies to:
cos θ * (sin θ + 1) / (sin² θ - 1)
Simplify further by recognizing that sin² θ - 1 is equal to -cos² θ (using the Pythagorean identity).
The final simplified expression is:
-cos θ