Home » » 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) $$\frac{7}{144}$$

### 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, $$\cfrac{2 - \sec^2 \theta}{1 - \cot \theta} = \cfrac{2 - (\frac{13}{12})^2}{1 - (\frac{12}{5})}$$
$$\cfrac{2 - (\frac{169}{144})}{1 - (\frac{12}{5})} = \cfrac{\frac{119}{144}}{-\frac{7}{12}} = \cfrac{119}{144} \times \cfrac{-12}{7} = \cfrac{-17}{12}$$

## 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, $$\cfrac{2 - \sec^2 \theta}{1 - \cot \theta} = \cfrac{2 - (\frac{13}{12})^2}{1 - (\frac{12}{5})}$$
$$\cfrac{2 - (\frac{169}{144})}{1 - (\frac{12}{5})} = \cfrac{\frac{119}{144}}{-\frac{7}{12}} = \cfrac{119}{144} \times \cfrac{-12}{7} = \cfrac{-17}{12}$$