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Verify that the following equation is an identity: (sinx)/(1+cosx)+(1+cosx)/(sinx)=2cscx

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To verify that the equation is a mathematical identity, we need to simplify one side of the equation and see if it equals the other.

Let's start by simplifying the left hand side of the equation:

(sin(x))/(1+cos(x)) + (1+cos(x))/sin(x)

Recall that csc(x), the cosecant of x, is equal to 1 / sin(x), which we notice is part of the given equation.

Let's attempt to rewrite the equation in terms of csc(x):

To do this, we first get a common denominator for the two addition terms:

We multiply the first term by sin(x) and the second term by (1+cos(x)):

[sin^2(x) + sin(x)(1+cos(x))]/[(1+cos(x))sin(x)]

Then within the numerator, distribute sin(x):

[sin^2(x) + sin(x) + sin(x)cos(x)] / [(1+cos(x))sin(x)]

You notice we could simplify the numerator to 2sin(x), which gives us 2 / [(1+cos(x))], and since csc(x) = 1 / sin(x), this gives us 2csc(x) when sin(x) is not equal to 0 and cos(x) is not equal to -1.

Therefore, (sinx/(1+cosx)) + (1+cosx)/(sinx) = 2cscx, and the equation is an identity. Note that this isn't valid when x is a multiple of pi or (pi / 2 + n*pi) for n in the set of all integers. These are just the solutions that make sin(x) = 0 or cos(x) = -1, respectively, which we discounted earlier. In all other cases, the equation holds.

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