[Sin A + Sin 3A + Sin 5A + Sin 7A]÷

[Cos A + Cos 3A + Cos 5A + Cos 7A]
=Tan 4A

Do the following rewrites:

sin(x) = sin(4x - 3x) = sin(4x) cos(3x) - cos(4x) sin(3x)

sin(7x) = sin(4x + 3x) = sin(4x) cos(3x) + cos(4x) sin(3x)

sin(3x) = sin(4x - x) = sin(4x) cos(x) - cos(4x) sin(x)

sin(5x) = sin(4x + x) = sin(4x) cos(x) + cos(4x) sin(x)

cos(x) = cos(4x - 3x) = cos(4x) cos(3x) + sin(4x) sin(3x)

cos(7x) = cos(4x + 3x) = cos(4x) cos(3x) - sin(4x) sin(3x)

cos(3x) = cos(4x - x) = cos(4x) cos(x) + sin(4x) sin(x)

cos(5x) = cos(4x + x) = cos(4x) cos(x) - sin(4x) sin(x)

So in the numerator, we have

sin(x) + sin(3x) + sin(5x) + sin(7x)

= 2 sin(4x) cos(3x) + 2 sin(4x) cos(x)

= 2 sin(4x) (cos(3x) + cos(x))

In the denominator,

cos(x) + cos(3x) + cos(5x) + cos(7x)

= 2 cos(4x) cos(3x) + 2 cos(4x) cos(x)

= 2 cos(4x) (cos(3x) + cos(x))

So we have

(sin(x) + sin(3x) + sin(5x) + sin(7x)) / (cos(x) + cos(3x) + cos(5x) + cos(7x))

= (2 sin(4x) (cos(3x) + cos(x))) / (2 cos(4x) (cos(3x) + cos(x)))

= sin(4x) / cos(4x)

= tan(4x)

QED

i got 142 as my answer

step-by-step explanation: