Question:
cosec theta= xsquare minus ysquare divided by x square plus y square where x and y belongs to real numbers gives real theta if and only if
Answer:
Given, cosec θ = (x2 - y2 )/(x2 + y2 )
=> 1/sin θ = (x2 - y2 )/(x2 + y2 )
=> x2 + y2 = sin θ(x2 - y2 )
=> x2 + y2 = sin θ * x2 - sin θ * y2
=> x2 - sin θ * x2 = -y2 - sin θ * y2
=> x2 (1 - sin θ) = -y2 (1 + sin θ)
=> x2 (1 - sin θ) = -y2 (1 + sin θ)
=> x2 /y2 = -(1 + sin θ)/(1 - sin θ)
=> x/y = √{-(1 + sin θ)/(1 - sin θ)}
Now, x/y is real if
-(1 + sin θ)/(1 - sin θ) ≥ 0
=> (1 + sin θ)/(1 - sin θ) ≤ 0
=> 1 + sin θ ≤ 1 - sin θ
=> sin θ + sin θ ≤ 1 - 1
=> 2 * sin θ ≤ 0
=> sin θ ≤ 0
=> sin θ ≤ sin 0
=> θ ≤ 0
So, cosec θ = (x2 - y2 )/(x2 + y2 ) is real if θ ≤ 0
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