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How do you solve (cscx-cotx)^4(cscx+cotx)^4=1?

Answer 1

Zoe Austin

Infinite solutions

#(cscx-cotx)^4(cscx+cotx)^4=1#

#((cscx-cotx)(cscx+cotx))^4=1#

#(csc^2x-cot^2x)^4=1#

#csc^2x-cot^2x=+-1#

#csc^2x-cot^2x=1. :. csc^2x=1+cot^2x#, Infinite solutions

#csc^2x-cot^2x=-1, :. 1+cot^2x-cot^2x=-1#, no solution

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Answer 2

Oliver Atkinson

To solve the equation ((\csc x - \cot x)^4(\csc x + \cot x)^4 = 1), we can use the fact that if two numbers multiply to give 1, then each of those numbers must be either 1 or -1. So, we can set each factor equal to 1 or -1 and solve for (x).

For ((\csc x - \cot x)^4 = 1): This implies (\csc x - \cot x = \pm 1). Solve each case separately.
For ((\csc x + \cot x)^4 = 1): This implies (\csc x + \cot x = \pm 1). Solve each case separately.

After solving these cases, you will have multiple solutions for (x). These solutions will depend on the range of (x) you are interested in. Remember that trigonometric functions have periodic behavior, so solutions may repeat after a certain interval.

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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.