cot^2x+cscx-1=0

Question

find solutions with trigonometric identities 

Answer 1

The first step is to use trigonometric identities to simplify this expression. One of the most basic trig identities you will ever need to memorize is:

sin^2x + cos^2x = 1

Now, we can manipulate this to find a way to simplify cot^2x by dividing both sides by sin^2x to get this:

sin^2x/sin^2x + cos^2x/sin^2x = 1/sin^2x

1 + cot^2x = csc^2x

cot^2x = csc^2x - 1

Now, let's put this back into the original problem:

csc^2x - 1 + cscx - 1 = 0

csc^2x + cscx - 2 = 0

This looks crazy, but it's really just a quadratic equation. Let's simplify it by letting u = cscx. You just get:

u^2 + u - 2 = 0

Which can be factored into:

(u + 2)(u - 1) = 0

Therefore, this is true for the values of u:

u = -2, and u = 1

Since u = cscx, this means that:

cscx = -2, and cscx = 1

Or:

sinx = -1/2, and sinx = 1

Assuming your answer needs to be between 0 and 360 degrees, taking the arcsin of these values gives:

x = 210, 330 (for the first relation), and x = 90 (for the second relation) degrees (multiply by pi/180 to get radians).