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

Problem

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

Solution

1. Express in terms of sine and cosine by using the quotient identities tan(x)=sin(x)/cos(x) and cot(x)=cos(x)/sin(x)

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

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

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

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

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

4. Separate the fraction into a product of two distinct ratios.

1/(sin(x)*cos(x))=1/sin(x)⋅1/cos(x)

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

1/sin(x)⋅1/cos(x)=csc(x)*sec(x)

Final Answer

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

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