Question

Prove the following.
(secθ + tanθ) (1 – sinθ) = cosθ

Answer 1

\[\left( \sec\theta + \tan\theta \right)\left( 1 - \sin\theta \right)\]

\[ = \left( \frac{1}{\cos\theta} + \frac{\sin\theta}{\cos\theta} \right)\left( 1 - \sin\theta \right)\]

\[ = \left( \frac{1 + \sin\theta}{\cos\theta} \right)\left( 1 - \sin\theta \right)\]

\[ = \frac{1 - \sin^2 \theta}{\cos\theta}\]

\[ = \frac{\cos^2 \theta}{\cos\theta} \left( \sin^2 \theta + \cos^2 \theta = 1 \right)\]

\[ = \cos\theta\]