Verify the Identity (5sin(x)+5cos(x))^2=25+25sin(2x)

Problem

(5*sin(x)+5*cos(x))2=25+25*sin(2*x)

Solution

1. Expand the squared binomial on the left side of the equation using the pattern (a+b)2=a2+2*a*b+b2

(5*sin(x)+5*cos(x))2=(5*sin(x))2+2*(5*sin(x))*(5*cos(x))+(5*cos(x))2

2. Simplify each term by squaring the coefficients and multiplying the middle terms.

(5*sin(x))2+2*(5*sin(x))*(5*cos(x))+(5*cos(x))2=25*sin2(x)+50*sin(x)*cos(x)+25*cos2(x)

3. Factor out the common factor of 25 from the squared trigonometric terms to prepare for the Pythagorean identity.

25*(sin2(x)+cos2(x))+50*sin(x)*cos(x)

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

25*(1)+50*sin(x)*cos(x)

5. Rewrite the second term to isolate the double-angle identity for sine, which is sin(2*x)=2*sin(x)*cos(x)

25+25*(2*sin(x)*cos(x))

6. Substitute the double-angle identity into the expression.

25+25*sin(2*x)

Final Answer

(5*sin(x)+5*cos(x))2=25+25*sin(2*x)

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