Question

How do you find the exact value of `csc^2theta-cottheta-1=0` in the interval `0<=theta<360`?

Answer 1

`45^@, 90^@, 225^@, 270^@`

Explanation:

`csc^2 t - cot t - 1 = 0`
`1/(sin^2 t) - cos t/(sin t) = 1`
Use common denominator then cross multiply
`1 - cos t.sin t = sin^2 t`
`1 - sin^2 t - cos t.sin t = 0`
Replace `(1 - sin^2 t)` by `cos^2 t` -->
`cos^2 t - sin t cos t) = 0`
cos t(cos t - sin t) = 0

a. cos t = 0 --> 2 solution arcs -->
arc`t = 90^@` and arc `t = 270^@`
b . cos t - sin t = - (sin t - cos t) = 0
Use trig identity:
`sin a - cos a = - sqrt2cos (a + 45)`
Replace a by t -->
`cos t - sin t = - (sin t - cos t) = sqrt2cos (t + 45) = 0`
cos (t + 45) = 0 --> 2 solution arcs -->
c. t + 45 = 90 --> `t = 90 - 45 = 45^@`
d. t + 45 = 270 --> `t = 270 - 45 = 225^@`
Answers for (0, 360):
`45^@, 90^@, 225^@, 270^@`