How do you simplify $csc X + cot X = \frac{sin X}{1 + cos X}$ $csc X + cot X = sin X / (1+cos X)$ ?

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How do you simplify $csc X + cot X = \frac{sin X}{1 + cos X}$ $csc X + cot X = sin X / (1+cos X)$ ?

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Answer 1 · 2016-03-22T00:59:54

This is not a valid identity.

Explanation:

We can prove this is invalid by using a test value of $x = \frac{π}{4}$ $x=pi/4$ :

$csc (\frac{π}{4}) + cot (\frac{π}{4}) \neq \frac{sin (\frac{π}{4})}{1 + cos (\frac{π}{4})}$ $csc(pi/4)+cot(pi/4)!=sin(pi/4)/(1+cos(pi/4))$

$\sqrt{2} + 1 \neq \frac{\frac{1}{\sqrt{2}}}{1 + \frac{1}{\sqrt{2}}}$ $sqrt2+1!=(1/sqrt2)/(1+1/sqrt2)$

$\sqrt{2} + 1 \neq \frac{1}{\sqrt{2} (1 + \frac{1}{\sqrt{2}})}$ $sqrt2+1!=1/(sqrt2(1+1/sqrt2))$

$\sqrt{2} + 1 \neq \frac{1}{\sqrt{2} + 1}$ $sqrt2+1!=1/(sqrt2+1)$

In fact, as we can might see is happening here, these functions are actually reciprocals of one another: they only intersect when their values equal $1$ $1$ or $- 1$ $-1$ .

We can also prove these are not equal by attempting to simplify the functions:

$csc x + cot x \neq \frac{sin x}{1 + cos x}$ $cscx+cotx!=sinx/(1+cosx)$

$\frac{1}{sin x} + \frac{cos x}{sin x} \neq \frac{sin x}{1 + cos x}$ $1/sinx+cosx/sinx!=sinx/(1+cosx)$

$\frac{1 + cos x}{sin x} \neq \frac{sin x}{1 + cos x}$ $(1+cosx)/sinx!=sinx/(1+cosx)$

Indeed, these functions are reciprocals of one another so the identity is invalid.

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How do you simplify $csc X + cot X = \frac{sin X}{1 + cos X}$ $csc X + cot X = sin X / (1+cos X)$ ?

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Explanation:

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How do you simplify $csc X + cot X = \frac{sin X}{1 + cos X}$ $csc X + cot X = sin X / (1+cos X)$ ?