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

Problem

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

Solution

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

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

2. Multiply the sine terms in the second part of the expression.

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

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

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

4. Combine the fractions into a single expression over the common denominator.

(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 final form.

sec(x)

Final Answer

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

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