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

Problem

sin(x)*tan(x)+cos(x)=sec(x)

Solution

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

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

2. Multiply the terms to simplify the first part of the expression.

sin^2(x)/cos(x)+cos(x)

3. Find a common denominator by multiplying the second term by cos(x)/cos(x)

sin^2(x)/cos(x)+cos^2(x)/cos(x)

4. Combine the fractions over the single denominator cos(x)

(sin^2(x)+cos^2(x))/cos(x)

5. Apply the Pythagorean identity sin^2(x)+cos^2(x)=1 to simplify the numerator.

1/cos(x)

6. Use the reciprocal identity 1/cos(x)=sec(x) to reach the final form.

sec(x)

Final Answer

sin(x)*tan(x)+cos(x)=sec(x)

Want more problems? Check here!