Discussion
If \(\cos X = 3/5\), then the value of \(3 + 3\tan^2 X\) is:
Answer: B
Correct answer: 25/3
Factor the 3 out and use a Pythagorean identity rather than finding \(\tan X\) first.
\[3 + 3\tan^2 X = 3\left(1 + \tan^2 X\right) = 3\sec^2 X\]
Since \(\cos X = \tfrac{3}{5}\), we have \(\sec X = \tfrac{5}{3}\), so
\[3\sec^2 X = 3 \times \frac{25}{9} = \frac{25}{3}\]
The same result follows the long way: \(\cos X = \tfrac{3}{5}\) gives a 3–4–5 triangle with \(\tan X = \tfrac{4}{3}\), and \(3 + 3\left(\tfrac{16}{9}\right) = 3 + \tfrac{16}{3} = \tfrac{25}{3}\). Stopping at \(\tfrac{16}{3}\) omits the leading 3.
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