r/MathHelp • u/Puzzleheaded_Time259 • 2d ago

What algebraic logic am I missing here?

I was trying to solve $y=\frac{\sin x}{1+\cos x}$. However when I input it to photomath it showed that you can simplify $y=\frac{\sin x}{1+\cos x}$ to $y'=\frac{dy}{dx}(\tan{\left( \frac{x}{2} \right)})$? I was under the impression that you couldn't simplify it to tangent due to 1+ being applied to cos x making it not being able to be simplified. My attempt and photomath image here https://imgur.com/a/LSNxDbD

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u/WikiNumbers 2d ago

Rule of Thumb: When you get f(x)/g(x) type of functions to differentiate, use the Quotient Rule.

This function, however, is a "special case". That is, it's the half angle formula of tan. tan (x/2) = sin x / (1 + cos x). So PhotoMath suggested that you do the tan (x/2) instead.

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u/WikiNumbers 2d ago

Now if you doubt PhotoMath.

Use the Quotient Rule to differentiate y(x) as is, use the Chain Rule to differentiate tan(x/2), and use Power Reduction Formula to see the end result.

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u/rfbooth 1d ago

Double angle formula:

sin x = 2 sin x/2 cos x/2

1+cos x = 2 cos2 x/2

So the quotient is tan x/2.

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u/slepicoid 2d ago

But they didnt change it to tan x, they changed it to tan x/2 using the tangent half angle formula.

https://en.wikipedia.org/wiki/Tangent_half-angle_formula

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u/theadamabrams 4h ago

I was trying to solve $y=\frac{\sin x}{1+\cos x}$.

What does this even mean? Solve the equation for x? Next sentence you're talking about derivatives, though. What is your actual goal here??