Verify the Identity tan(x)^2+1=sec(x)^2

Problem

tan2(x)+1=sec2(x)

Solution

1. Recall the fundamental definitions of the trigonometric functions in terms of sine and cosine.

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

sec(x)=1/cos(x)

2. Substitute the definition of tangent into the left side of the identity.

tan2(x)+1=(sin(x)/cos(x))2+1

3. Distribute the exponent to the numerator and denominator.

tan2(x)+1=sin2(x)/cos2(x)+1

4. Find a common denominator to combine the terms.

tan2(x)+1=sin2(x)/cos2(x)+cos2(x)/cos2(x)

5. Combine the fractions over the common denominator.

tan2(x)+1=(sin2(x)+cos2(x))/cos2(x)

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

tan2(x)+1=1/cos2(x)

7. Recognize that the resulting expression is the definition of the squared secant function.

1/cos2(x)=sec2(x)

Final Answer

tan2(x)+1=sec2(x)

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