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

Problem

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

Solution

1. Rewrite the trigonometric functions inside the parentheses in terms of sine and cosine using the quotient identities.

cot(x)=cos(x)/sin(x)

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

2. Substitute these expressions back into the left side of the equation.

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

3. Find a common denominator for the terms inside the parentheses to combine the fractions.

cos(x)/sin(x)+sin(x)/cos(x)=(cos2(x)+sin2(x))/(sin(x)*cos(x))

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

1/(sin(x)*cos(x))

5. Distribute the sin(x) term outside the parentheses into the simplified fraction.

sin(x)⋅1/(sin(x)*cos(x))

6. Cancel the common factor of sin(x) from the numerator and denominator.

1/cos(x)

7. Apply the reciprocal identity to reach the final form.

1/cos(x)=sec(x)

Final Answer

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

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