Multiple choice

In any right angled triangle. Find the value of $\cot^{2}\theta - \dfrac {1}{\sin^{2}\theta}$,where $\theta$ is one of the angles .

  1. $1$
  2. $0$
  3. $-1$

  4. none of these
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C Correct answer
Explanation

cot^2 theta - 1/sin^2 theta = cot^2 theta - cosec^2 theta. Since cosec^2 theta - cot^2 theta = 1, then cot^2 theta - cosec^2 theta = -1.

AI explanation

Using the trigonometric identity cosecant squared theta equals one plus cotangent squared theta, we can substitute one over sine squared theta with cosecant squared theta in the given expression. This changes the expression to cotangent squared theta minus cosecant squared theta. Rearranging the identity gives cotangent squared theta minus cosecant squared theta equals negative one. The value of the expression is negative one.

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