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

Problem

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

Solution

1. Rewrite the left side of the equation using the fundamental trigonometric identities for cotangent and tangent in terms of sine and cosine.

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

2. Find a common denominator to combine the two fractions, which is sin(x)*cos(x)

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

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

(cos2(x)+sin2(x))/(sin(x)*cos(x))=1/(sin(x)*cos(x))

4. Separate the fraction into a product of two distinct reciprocal functions.

1/(sin(x)*cos(x))=1/cos(x)⋅1/sin(x)

5. Substitute the reciprocal identities sec(x)=1/cos(x) and csc(x)=1/sin(x) to reach the final form.

1/cos(x)⋅1/sin(x)=sec(x)*csc(x)

Final Answer

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

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