G)
(Sin 5A - Sin 7A - Sin 4A + Sin 8A)÷
(Cos 4A - Cos 5A - Cos 8A + Cos 7A)=
Cot 6A

We use a similar strategy as in your previous question. Rewrite:

sin(5x) = sin(6x - x) = sin(6x) cos(x) - cos(6x) sin(x)

sin(7x) = sin(6x + x) = sin(6x) cos(x) + cos(6x) sin(x)

→ sin(5x) - sin(7x) = -2 cos(6x) sin(x)

sin(4x) = sin(6x - 2x) = sin(6x) cos(2x) - cos(6x) sin(2x)

sin(8x) = sin(6x + 2x) = sin(6x) cos(2x) + cos(6x) sin(2x)

→   sin(8x) - sin(4x) = 2 cos(6x) sin(2x)

cos(5x) = cos(6x - x) = cos(6x) cos(x) + sin(6x) sin(x)

cos(7x) = cos(6x + x) = cos(6x) cos(x) - sin(6x) sin(x)

→   cos(7x) - cos(5x) = -2 sin(6x) sin(x)

cos(4x) = cos(6x - 2x) = cos(6x) cos(2x) + sin(6x) sin(2x)

cos(8x) = cos(6x + 2x) = cos(6x) cos(2x) - sin(6x) sin(2x)

→   cos(4x) - cos(8x) = 2 sin(6x) sin(2x)

Then

(sin(5x) - sin(7x) - sin(4x) + sin(8x)) / (cos(4x) - cos(5x) - cos(8x) + cos(7x))

= (2 cos(6x) sin(2x) - 2 cos(6x) sin(x)) / (2 sin(6x) sin(2x) - 2 sin(6x) sin(x))

= (2 cos(6x) (sin(2x) - sin(x))) / (2 sin(6x) (sin(2x) - sin(x)))

= cos(6x) / sin(6x)

= cot(6x)

QED

practice. practice. practice.

the answer is 1 becuz if it were 1.0 that would rip apart the entire so ya. : )

~potato.

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