Multiple choice

The equation: $\sin ^{ 2 }{ \theta } -\cfrac { 4 }{ \sin ^{ 2 }{ \theta } -1 } =1-\cfrac { 4 }{ \sin ^{ 2 }{ \theta } -1 } $ has:

  1. no root

  2. one root

  3. two root

  4. infinite roots
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation is sin^2(theta) - 4/(sin^2(theta) - 1) = 1 - 4/(sin^2(theta) - 1). Subtracting the common term from both sides gives sin^2(theta) = 1. This implies sin^2(theta) - 1 = 0, which makes the denominator zero. Thus, there is no valid root.

AI explanation

By canceling the common term $-4/(\sin^2 \theta - 1)$ from both sides of the equation, we are left with $\sin^2 \theta = 1$. Solving this gives $\sin \theta = \pm 1$, which would make the denominator $\sin^2 \theta - 1$ equal to zero. Because division by zero is undefined, any solutions derived from $\sin^2 \theta = 1$ are extraneous. Therefore, the equation has no root.

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