SOLUTION: What are the x's of: ((cot^2)x) +(csc^2)x)) = 7

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Question 747134: What are the x's of: ((cot^2)x) +(csc^2)x)) = 7
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
cot%28x%29=cos%28x%29%2Fsin%28x%29 and csc%28x%29=1%2Fsin%28x%29 so
%28cot%28x%29%29%5E2%2B%28csc%28x%29%29%5E2=7 --> %28cos%28x%29%29%5E2%2F%28sin%28x%29%29%5E2%2B1%2F%28sin%28x%29%29%5E2=7
Multiplying both sides times %28sin%28x%29%29%5E2 we can eliminate denominators to get
%28cos%28x%29%29%5E2%2B1=7%28sin%28x%29%29%5E2 --> 1-%28sin%28x%29%29%5E2%2B1=7%28sin%28x%29%29%5E2 --> 2-%28sin%28x%29%29%5E2=7%28sin%28x%29%29%5E2 --> 2=8%28sin%28x%29%29%5E2 --> 2%2F8=%28sin%28x%29%29%5E2 --> %28sin%28x%29%29%5E2=1%2F4
The solutions must yield sin%28x%29=1%2F2 or sin%28x%29=-1%2F2
Looking for solution between 0%5Eo and 360%5Eo (between 0 and 2pi radians),
sin%28x%29=1%2F2 happens for
x=30%5Eo (pi%2F6radians) and x=150%5Eo (5pi%2F6radians)
and sin%28x%29=-1%2F2 happens for
x=210%5Eo (7pi%2F6radians) and x=330%5Eo (11pi%2F6radians).
All solutions can be expressed as
x=k%2A180%5Eo+%2B-+30%5Eo or x=k%2Api+%2B-+pi%2F6