Verify the Identity sin(x)^2+cos(x)^2=1

Problem

sin2(x)+cos2(x)=1

Solution

1. Identify the definitions of the trigonometric functions using a right triangle with angle x opposite side a adjacent side b and hypotenuse c

2. Substitute the ratios into the expression using sin(x)=a/c and cos(x)=b/c

sin2(x)+cos2(x)=(a/c)2+(b/c)2

3. Simplify the fractions by squaring the numerators and denominators.

sin2(x)+cos2(x)=(a2)/(c2)+(b2)/(c2)

4. Combine the fractions over a common denominator.

sin2(x)+cos2(x)=(a2+b2)/(c2)

5. Apply the Pythagorean theorem, which states that for any right triangle, a2+b2=c2

sin2(x)+cos2(x)=(c2)/(c2)

6. Divide the terms to reach the final value.

sin2(x)+cos2(x)=1

Final Answer

sin2(x)+cos2(x)=1

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