Simplify

Problem

cos(x)/(sin(x)−1)⋅(sin(x)+1)/(sin(x)+1)

Solution

1. Multiply the numerators and denominators to combine the expression into a single fraction.

(cos(x)*(sin(x)+1))/((sin(x)−1)*(sin(x)+1))

2. Expand the denominator using the difference of squares formula, (a−b)*(a+b)=a^2−b^2

(cos(x)*(sin(x)+1))/(sin^2(x)−1)

3. Apply a Pythagorean identity to rewrite the denominator. Since sin^2(x)+cos^2(x)=1 it follows that sin^2(x)−1=−cos^2(x)

(cos(x)*(sin(x)+1))/(−cos^2(x))

4. Simplify the fraction by canceling a common factor of cos(x) from the numerator and the denominator.

(sin(x)+1)/(−cos(x))

5. Distribute the negative sign to express the final simplified form.

−(sin(x)+1)/cos(x)

Final Answer

cos(x)/(sin(x)−1)⋅(sin(x)+1)/(sin(x)+1)=−(sin(x)+1)/cos(x)

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