Victor Jenkins
Dec 07, 2024
(1 – 3cosx – 4cos^2x) / sin^2x = (1 – 4cosx)/ (1 – cosx)?
How can I manipulate the expression ((1 - 3\cos x - 4\cos^2 x) / \sin^2 x) to show that it is equal to ((1 - 4\cos x) / (1 - \cos x)) without altering the right side of the equation?
2 Answers
(1 – 3cosx – 4cos^2x) / sin^2x =
(1+cos x)(1-4cos x) /(1-cos^2 x) =
(1+cos x)(1-4cos x) /[(1-cos x)(1+cos x)] =
(1-4cos x) /(1-cos x)
You can change both sides, you just can’t cross the equals sign. That aside, you don’t need to change the right in this case, factor the numerator like a quadratic:
(1 + cos(x))(1 – 4cos(x)) / sin²(x) = RHS
Use the pythagorean theorem to change sin²(x):
(1+cos(x))(1-4cos(x)) / (1 – cos²(x)) = RHS
Factor:
(1+cos(x))(1 – 4cos(x)) / (1 + cos(x))(1 – cos(x)) = RHS
Cancel:
(1 – 4cos(x)) / (1 – cos(x)) = RHS
QED
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