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How do you prove #Csc(x) - Cos(x)Cot(x) = Sin(x)#?

Answer 1

Asher Friedman

see below

Left Side:#=1/sinx - (cosx cosx/sinx)#

#=1/sinx -cos^2x/sinx#

#=(1-cos^2x)/sinx#

#=sin^2x/sin x#

#=sinx#

#=#Right Side

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Answer 2

Kai Anderson

To prove that csc(x) - cos(x)cot(x) = sin(x), we can start with the left-hand side of the equation and use trigonometric identities to simplify it step by step.

Starting with:

csc(x) - cos(x)cot(x)

We know that csc(x) = 1/sin(x) and cot(x) = cos(x)/sin(x). Substituting these into the expression, we get:

1/sin(x) - cos(x)(cos(x)/sin(x))

Next, we simplify:

= 1/sin(x) - cos^2(x)/sin(x)

Now, we find a common denominator:

= (1 - cos^2(x))/sin(x)

We recognize that 1 - cos^2(x) = sin^2(x) (using the Pythagorean identity). Substituting this into the expression, we get:

= sin^2(x)/sin(x)

= sin(x)

Therefore, csc(x) - cos(x)cot(x) simplifies to sin(x), which completes the proof.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.