Verify the Identity

Problem

cos(x)−cos(x)/(1−tan(x))=(sin(x)*cos(x))/(sin(x)−cos(x))

Solution

1. Express the tangent function in terms of sine and cosine using the quotient identity tan(x)=sin(x)/cos(x)

cos(x)−cos(x)/(1−sin(x)/cos(x))

2. Simplify the denominator of the fraction by finding a common denominator of cos(x)

cos(x)−cos(x)/(cos(x)−sin(x))/cos(x)

3. Multiply the numerator by the reciprocal of the denominator to simplify the complex fraction.

cos(x)−cos2(x)/(cos(x)−sin(x))

4. Find a common denominator for the entire expression, which is cos(x)−sin(x)

(cos(x)*(cos(x)−sin(x))−cos2(x))/(cos(x)−sin(x))

5. Distribute the cos(x) in the numerator.

(cos2(x)−sin(x)*cos(x)−cos2(x))/(cos(x)−sin(x))

6. Combine like terms in the numerator by canceling cos2(x) and −cos2(x)

(−sin(x)*cos(x))/(cos(x)−sin(x))

7. Factor out a −1 from the denominator to match the target expression.

(−sin(x)*cos(x))/(−(sin(x)−cos(x)))

8. Cancel the negative signs to reach the final form.

(sin(x)*cos(x))/(sin(x)−cos(x))

Final Answer

cos(x)−cos(x)/(1−tan(x))=(sin(x)*cos(x))/(sin(x)−cos(x))

Want more problems? Check here!