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

Problem

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

Solution

1. Use the quotient identity to rewrite the tangent function in terms of sine and cosine.

tan(x)=sin(x)/cos(x)

2. Substitute this identity back into the original expression.

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

3. Multiply the terms in the second part of the expression.

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

4. Find a common denominator to combine the terms into a single fraction.

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

5. Combine the numerators over the common denominator.

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

6. Apply the Pythagorean identity where the sum of the squares of sine and cosine equals one.

cos2(x)+sin2(x)=1

7. Simplify the resulting fraction.

1/cos(x)

8. Use the reciprocal identity to express the final result as a single trigonometric function.

1/cos(x)=sec(x)

Final Answer

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

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