Question

If cot theta = b/a, then the value of
cos theta + sin theta/cos theta - sin theta

Answer 1

It's equal to $\frac{b + \sqrt{ {a}^{2} - {b}^{2} } }{b - { \sqrt{ {a}^{2} - {b}^{2} } }^{2} }$

${sin}^{2} \alpha + {cos}^{2} \alpha = 1 \\ {sin}^{2} \alpha = 1 - {cos}^{2} \alpha \\ {sin}^{2} \alpha = 1 - \frac{ {b}^{2} }{ {a}^{2} } \\ sin \alpha = \frac{ \sqrt{ {a}^{2} - {b}^{2} } }{a}$

Thus,

$\frac{cos \alpha + sin \alpha }{cos \alpha - sin \alpha } \\ = \frac{ \frac{b}{a} + \frac{ \sqrt{ {a}^{2} - {b}^{2} } }{a} }{ \frac{b}{a} - \frac{ \sqrt{ {a}^{2} - {b}^{2} } }{a} } \\ = \frac{b + \sqrt{ {a}^{2} - {b}^{2} } }{b - \sqrt{ {a}^{2} - {b}^{2} } }$

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