Verify the Identity (tan(x)-sin(-x))/(1+cos(x))=tan(x)

Problem

(tan(x)−sin(−x))/(1+cos(x))=tan(x)

Solution

1. Apply the odd-even identity for the sine function, which states that sin(−x)=−sin(x)

sin(−x)=−sin(x)

2. Substitute this identity into the numerator of the left-hand side.

(tan(x)−(−sin(x)))/(1+cos(x))

3. Simplify the signs in the numerator.

(tan(x)+sin(x))/(1+cos(x))

4. Rewrite the tangent function using the quotient identity tan(x)=sin(x)/cos(x)

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

5. Factor out sin(x) from the numerator.

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

6. Find a common denominator for the terms inside the parentheses in the numerator.

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

7. Cancel the common factor of 1+cos(x) from the numerator and the denominator.

sin(x)/cos(x)

8. Apply the quotient identity to simplify the remaining expression back to tan(x)

tan(x)

Final Answer

(tan(x)−sin(−x))/(1+cos(x))=tan(x)

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