Can anyone help prove this trigonometric identity?

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Can anyone help prove this trigonometric identity?

$\frac{sin x}{1 + cos x} + \frac{cos x}{sin x} = csc x$ $sinx/(1+cos x) + cos x/sin x = csc x$

2 Answers

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Answer 1 · 2016-11-29T09:02:29

Cross multiply and simplify LHS (as below)

Explanation:

To prove: $\frac{sin x}{1 + cos x} + \frac{cos x}{sin x} = csc x$ $sinx/(1+cosx) + cosx/sinx = cscx$

$L H S = \frac{sin x}{1 + cos x} + \frac{cos x}{sin x}$ $LHS = sinx/(1+cosx) + cosx/sinx$

$= \frac{{sin}^{2} x + cos x + {cos}^{2} x}{sin x + sin x cos x}$ $=(sin^2x+cosx+cos^2x)/(sinx+sinxcosx)$

Since: ${sin}^{2} x + {cos}^{2} x = 1$ $sin^2x + cos^2x=1$

$L H S = \frac{1 + cos x}{sin x (1 + cos x)}$ $LHS= (1+cosx)/(sinx(1+cosx))$

$= cancel(1+cosx)*1/(sinx*cancel(1+cosx))$

$=1/sinx = cscx = RHS$

Answer 2 · 2016-11-29T13:23:12

$sinx/(1+cosx)+cosx/sinx = ((sinx))/((1+cosx)) * ((1-cosx))/((1-cosx))+cosx/sinx$

$= (sinx(1-cosx))/(1-cos^2x) +cosx/sinx$

$= (sinx(1-cosx))/(sin^2x) +cosx/sinx$

$= (1-cosx)/sinx +cosx/sinx$

$= (1-cosx+cosx)/sinx$

$= 1/sinx$

$= cscx$

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Can anyone help prove this trigonometric identity?

$\frac{sin x}{1 + cos x} + \frac{cos x}{sin x} = csc x$ $sinx/(1+cos x) + cos x/sin x = csc x$

2 Answers

Explanation:

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Impact of this question

Science

Math

Humanities

... and beyond

Can anyone help prove this trigonometric identity?

sinx/(1+cos x) + cos x/sin x = csc x sinx1+cosx+cosxsinx=cscxsinx/(1+cos x) + cos x/sin x = csc x

2 Answers

Explanation:

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Impact of this question

$\frac{sin x}{1 + cos x} + \frac{cos x}{sin x} = csc x$ $sinx/(1+cos x) + cos x/sin x = csc x$