Simplify (sec(x)^2-1)/(sec(x)^2)

Problem

(sec^2(x)−1)/sec^2(x)

Solution

1. Identify the trigonometric identity for the numerator using the Pythagorean identity tan^2(x)+1=sec^2(x)

2. Substitute tan^2(x) for sec^2(x)−1 in the expression.

tan^2(x)/sec^2(x)

3. Rewrite the trigonometric functions in terms of sine and cosine using the definitions tan(x)=sin(x)/cos(x) and sec(x)=1/cos(x)

sin^2(x)/cos^2(x)/1/cos^2(x)

4. Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator.

sin^2(x)/cos^2(x)⋅cos^2(x)

5. Cancel the common factor of cos^2(x) to find the final simplified form.

sin^2(x)

Final Answer

(sec^2(x)−1)/sec^2(x)=sin^2(x)

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