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

Problem

(sin(x)+cos(x))^2=1+sin(2*x)

Solution

1. Expand the left side of the equation using the square of a binomial formula (a+b)^2=a^2+2*a*b+b^2

sin^2(x)+2*sin(x)*cos(x)+cos^2(x)

2. Rearrange the terms to group the squared trigonometric functions together.

sin^2(x)+cos^2(x)+2*sin(x)*cos(x)

3. Apply the Pythagorean identity sin^2(x)+cos^2(x)=1 to simplify the first part of the expression.

1+2*sin(x)*cos(x)

4. Apply the double-angle identity for sine, which states that sin(2*x)=2*sin(x)*cos(x)

1+sin(2*x)

Final Answer

(sin(x)+cos(x))^2=1+sin(2*x)

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