I need help with trigonometric identities and verifying them help solve (sin x + cos x) squared = sin 2 x +1?

Question

I have a final coming up and these are my weakness. Pleas help me keep my gpa of 4.0

Answer 1

(sin x+ cos x)^2 = sin^2x + 2sin x cos x + cos^2 x using (a+b)^2
= sin^2 x + cos ^2 x + 2 sinx cos x
= 1 + sin 2x
becuase sin^2x+ cos^2x =1
and sin 2x = sin(x+x) = sin x cos x + cosx sin x = 2sin xcos x
using the formula sin(A+B) = sin A cos B + cos A sin B

Answer 2

Consider the right-hand side (RHS) of the given equation; i.e. sin(2x) + 1. We have to show that this expression must be equal to the left-hand side (LHS); i.e., [sin(x) + cos(x)]^2.

That is,
RHS = sin(2x) + 1.
     = 2 sin(x) cos(x) + 1, by double angle identity of sine function
     = 2 sin(x) cos(x) + [sin(x)]^2 + [cos(x)]^2, by applying the Pythagorean identity
     = [sin(x)]^2 + 2 sin(x) cos(x) + [cos(x)]^2, by associative law of addition
     = [sin(x) + cos(x)]^2, by factoring
     = LHS.

Therefore, we have shown that [sin(x) + cos(x)]^2 = sin(2x) + 1.

Answer 3

that is now not honestly. attempt A = B = C = pi/3. Then the LHS is sin(2pi/3) = sqrt(3) /2 and the RHS is two*sin^2(pi/3)*cos(pi/3) = 2*(sqrt(3) /2)^2 *(a million/2) = 3/4. i ended up with sin(2A) - sin(2B) + sin(2C) = 4*cos(A)*sin(B)*cos(C), which does artwork with the above party suggestion. i will attend to take heed to from you ideal to the accuracy of the given fact contained in the previous spending the time writing out an information.

Answer 4

(sinx + cosx)^2 = sin2x + 1

sin^2 x + cos^2 x + 2cosxsinx = sin2x + 1

Remember : sin^2 x + cos^2 x = 1

and

2cosxsinx = sin2x


so plug in those two identities and you get

(1) + (sin2x) = sin2x + 1

Left = Right!

Answer 5

(sin(x) + xos(x)^2 = sin^2(x) + 2sin(x)cos(x) +cos^2(x)

sin^2(x) + cos^2(x) = 1 (a well-known identity), so

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

Also, the double angle formula for sin is sin(2x)=2sin(x)cos(x); substituting you get

1+sin(2x)