Discussion
If \(\sin\theta = \dfrac{x}{y}\), then find the value of \(2.\tan^{2}\theta\)?
Answer: C
The value is \(\dfrac{2x^{2}}{y^{2} - x^{2}}\).
From \(\sin\theta = x/y\), treat \(x\) as the opposite side and \(y\) as the hypotenuse. The adjacent side is then \(\sqrt{y^{2} - x^{2}}\):
\[ \begin{aligned} \tan\theta &= \frac{x}{\sqrt{y^{2} - x^{2}}} \\ 2\tan^{2}\theta &= \frac{2x^{2}}{y^{2} - x^{2}} \end{aligned} \]
The stem's '2.' is a multiplication dot. Options A and B are \(\csc\theta\) and \(\sin\theta\) themselves, and option D has the wrong power and the wrong sign in the denominator.
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