Simplify cos(x)+tan(x)*sin(x)

Problem

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

Solution

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

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

2. Multiply the terms in the second part of the expression to combine the sine functions.

cos(x)+sin2(x)/cos(x)

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

cos2(x)/cos(x)+sin2(x)/cos(x)

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

(cos2(x)+sin2(x))/cos(x)

5. Apply the Pythagorean identity cos2(x)+sin2(x)=1 to simplify the numerator.

1/cos(x)

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

sec(x)

Final Answer

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

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