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

Problem

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

2. Multiply the sine terms in the first part of the expression to simplify the numerator.

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

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

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

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

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

5. Apply the Pythagorean identity sin2(x)+cos2(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

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

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