Verify the Identity sec(x)^2csc(x)^2=sec(x)^2+csc(x)^2

Problem

sec2(x)*csc2(x)=sec2(x)+csc2(x)

Solution

1. Express the right side of the identity in terms of sine and cosine using the reciprocal identities sec(x)=1/cos(x) and csc(x)=1/sin(x)

sec2(x)+csc2(x)=1/cos2(x)+1/sin2(x)

2. Find a common denominator to add the two fractions. The common denominator is cos2(x)*sin2(x)

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

3. Combine the fractions into a single expression over the common denominator.

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

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

1/(cos2(x)*sin2(x))

5. Separate the fraction back into a product of two reciprocal functions.

1/cos2(x)⋅1/sin2(x)

6. Substitute the reciprocal identities back to reach the form of the left side.

sec2(x)*csc2(x)

Final Answer

sec2(x)*csc2(x)=sec2(x)+csc2(x)

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