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

Problem

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

Solution

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

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

2. Find a common denominator for the terms in the numerator to combine them into a single fraction.

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

3. Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator.

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

4. Cancel the common factor (sin(x)+cos(x)) from the numerator and the denominator.

1/cos(x)

5. Apply the reciprocal identity 1/cos(x)=sec(x) to reach the right side of the identity.

sec(x)

Final Answer

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

Want more problems? Check here!