Csc^2x-cotx-1=0 (algebra/trig)?
I got up to:
(1/sinx)^2 - (cosx/sinx) - 1 = 0
I feel like I would know what to do next if that stupid 1 wasn't there. I thought you would put the 1 over 1 and times it by sinx bc it's the common denominator but I'm not sure >.<
Update:And if you help me find the exact value of x in the interval 0 degrees less than or equal to x less than 360 degrees, that would be nice :)
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Why'd you do all that mess?
csc(x)^2 - cot(x)^2 = 1
csc(x)^2 - cot(x) - 1 = 0
csc(x)^2 - 1 - cot(x) = 0
cot(x)^2 - cot(x) = 0
cot(x) * (cot(x) - 1) = 0
cot(x) = 0
cos(x) / sin(x) = 0
cos(x) = 0
x = pi/2 , 3pi/2
x = 90 degrees, 270 degrees
cot(x) - 1 = 0
cot(x) = 1
tan(x) = 1/1
tan(x) = 1
x = 45 , 225
x = 45 , 90 , 225 , 270
Substitute 1 + cot^2(x) for cosec^2(x):
1 + cot^2(x) - cot(x) - 1 = 0
Combine the like terms:
cot^2(x) - cot(x) = 0
Rewrite the reciprocal ratios as their normal ratios:
1/tan^2(x) - 1/tan(x) = 0
Multiply everything by tan^2(x):
1 - tan(x) = 0
Add tan(x) to both sides:
tan(x) = 1
As the result is positive we know that one solution must lie within the first quadrant and the other must lie within the third quadrant, solve for each:
x = 45°, 225°
You can use (or use the equation to find) the identity csc^2x = 1 + cot^2x to yield
cot^2x - cotx = 0
And then finding solutions becomes easier.