Question

Sinx+sin2x+sin3x/(cosx+cos2x+cos3x)=tg2x

Answer 1

$\frac{sinx+sin2x+sin3x}{cosx+cos2x+cos3x}=tg2x\\\\sinx+sin2x+sin3x=sin(2x-x)+sin2x+sin(2x+x)=\\sin2x+sin2xcosx-cos2xsinx+sin2xcosx+cos2xsinx=\\sin2x+2sin2xcosx=sin2x(1+2cosx);\\\\cosx+cos2x+cos3x=cos(2x-x)+cos2x+cos(2x+x)=\\cos2x+cos2xcosx+sin2xsinx+cos2xcosx-sin2xsinx=\\cos2x+2cos2xcosx=cos2x(1+2cosx);\\\\\frac{sin2x(1+cosx)}{cos2x(1+cos2x)}=\frac{sin2x}{cos2x}=tg2x$

Answer 2

To prove that the given equation is true, we'll work on simplifying the left side of the equation and show that it equals the right side.

Starting with the left side of the equation:

sin(x) + sin(2x) + sin(3x) / (cos(x) + cos(2x) + cos(3x))

To simplify this expression, we'll express the numerator and denominator in terms of sines and cosines.

The numerator can be rewritten using the sum-to-product identities:

sin(x) + sin(2x) + sin(3x) = 2sin(x)cos(x) + 2sin(x)cos^2(x) + 2sin(x)cos^2(x) - 2sin^3(x)

The denominator can also be rewritten using the sum-to-product identities:

cos(x) + cos(2x) + cos(3x) = 2cos^2(x/2) - 1 + 2cos^2(x) - 1 + 2cos^2(3x/2) - 1

Combining the numerator and denominator:

(2sin(x)cos(x) + 2sin(x)cos^2(x) + 2sin(x)cos^2(x) - 2sin^3(x)) / (2cos^2(x/2) - 1 + 2cos^2(x) - 1 + 2cos^2(3x/2) - 1)

Now, let's simplify the expression further:

2sin(x)(cos(x) + 2cos^2(x) + 2cos^2(x) - 2sin^2(x)) / (2cos^2(x/2) + 2cos^2(x) + 2cos^2(3x/2) - 2)

Simplifying the numerator:

2sin(x)(cos(x) + 4cos^2(x) - 2sin^2(x)) / (2cos^2(x/2) + 2cos^2(x) + 2cos^2(3x/2) - 2)

Using the Pythagorean identity cos^2(x) = 1 - sin^2(x):

2sin(x)(cos(x) + 4(1 - sin^2(x)) - 2sin^2(x)) / (2cos^2(x/2) + 2cos^2(x) + 2cos^2(3x/2) - 2)

2sin(x)(cos(x) + 4 - 4sin^2(x) - 2sin^2(x)) / (2cos^2(x/2) + 2cos^2(x) + 2cos^2(3x/2) - 2)

2sin(x)(4cos(x) - 6sin^2(x) + 4) / (2cos^2(x/2) + 2cos^2(x) + 2cos^2(3x/2) - 2)

Now, let's simplify the denominator:

2cos^2(x/2) + 2cos^2(x) + 2cos^2(3x/2) - 2

Using the double-angle formula cos(2θ) = 2cos^2(θ) - 1:

2(cos^2(x/2) + 1) + 2(cos^2(x) + 1) + 2(cos^2(3x/2) + 1) - 2

2cos^2(x/2) + 2 + 2cos^2(x