Simplify (sec(x))/(csc(x))+(csc(x))/(sec(x))

Problem

sec(x)/csc(x)+csc(x)/sec(x)

Solution

1. Rewrite the trigonometric functions in terms of sine and cosine using the reciprocal identities sec(x)=1/cos(x) and csc(x)=1/sin(x)

1/cos(x)/1/sin(x)+1/sin(x)/1/cos(x)

2. Simplify each complex fraction by multiplying by the reciprocal of the denominator.

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

3. Apply the quotient identities tan(x)=sin(x)/cos(x) and cot(x)=cos(x)/sin(x) to express the terms as tangent and cotangent.

tan(x)+cot(x)

4. Find a common denominator to combine the fractions from step 2.

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

5. Use the Pythagorean identity sin2(x)+cos2(x)=1 to simplify the numerator.

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

6. Apply reciprocal identities again to write the expression in terms of secant and cosecant.

sec(x)*csc(x)

Final Answer

sec(x)/csc(x)+csc(x)/sec(x)=sec(x)*csc(x)

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