Home » Past Questions » General Paper » Given that \(\sin \theta = \frac{5}{13}\), find the value of \(\cfrac{2 - \sec^2\theta}{1 + \cot \theta}\)...

Given that \(\sin \theta = \frac{5}{13}\), find the value of \(\cfrac{2 - \sec^2\theta}{1 + \cot \theta}\)...


Question

Given that \(\sin \theta = \frac{5}{13}\), find the value of \(\cfrac{2 - \sec^2\theta}{1 + \cot \theta}\)

Options

A) \(-\frac{17}{12}\)

B) \(\frac{144}{35}\)

C) \(-\frac{35}{144}\)

D) \(\frac{7}{144}\)

The correct answer is A.

Explanation:

If \(\sin \theta = \frac{5}{13}\), then \(\cos \theta = \frac{12}{13}\);
Recall \(5^2 + 12^2 = 13^2\)
\(\sec \theta = \frac{1}{\cos \theta} = \frac{13}{12}\) and \(\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{12}{5}\)
Therefore, \(\cfrac{2 - \sec^2 \theta}{1 - \cot \theta} = \cfrac{2 - (\frac{13}{12})^2}{1 - (\frac{12}{5})}\)
\(\cfrac{2 - (\frac{169}{144})}{1 - (\frac{12}{5})} = \cfrac{\frac{119}{144}}{-\frac{7}{12}} = \cfrac{119}{144} \times \cfrac{-12}{7} = \cfrac{-17}{12}\)

More Past Questions:


Discussion (2)

  • The answer is C you changed the sign of the denominator while solving

  • If \(\sin \theta = \frac{5}{13}\), then \(\cos \theta = \frac{12}{13}\);
    Recall \(5^2 + 12^2 = 13^2\)
    \(\sec \theta = \frac{1}{\cos \theta} = \frac{13}{12}\) and \(\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{12}{5}\)
    Therefore, \(\cfrac{2 - \sec^2 \theta}{1 - \cot \theta} = \cfrac{2 - (\frac{13}{12})^2}{1 - (\frac{12}{5})}\)
    \(\cfrac{2 - (\frac{169}{144})}{1 - (\frac{12}{5})} = \cfrac{\frac{119}{144}}{-\frac{7}{12}} = \cfrac{119}{144} \times \cfrac{-12}{7} = \cfrac{-17}{12}\)

    Reply
    Like