Question

SinA (1+TanA) + CosA (1+CotA) =

Answer 1

$$\sin \alpha (1 + \tan \alpha) + \cos \alpha (1 + \cot \alpha) = ?$$

Solution:

$$\begin{array}{l} \sin \alpha (1 + \tan \alpha) + \cos \alpha (1 + \cot \alpha) = \sin \alpha \left(1 + \frac {\sin \alpha}{\cos \alpha}\right) + \cos \alpha \left(1 + \frac {\cos \alpha}{\sin \alpha}\right) \\ = \sin \alpha \frac {\cos \alpha + \sin \alpha}{\cos \alpha} + \cos \alpha \frac {\sin \alpha + \cos \alpha}{\sin \alpha} \\ = (\cos \alpha + \sin \alpha) \left(\frac {\sin \alpha}{\cos \alpha} + \frac {\cos \alpha}{\sin \alpha}\right) \\ = (\cos \alpha + \sin \alpha) \left(\frac {\sin^ {2} \alpha + \cos^ {2} \alpha}{\cos \alpha * \sin \alpha}\right) = \frac {\cos \alpha + \sin \alpha}{\cos \alpha * \sin \alpha} = \frac {1}{\sin \alpha} + \frac {1}{\cos \alpha} \\ \end{array}$$

Answer: $\frac{1}{\sin\alpha} + \frac{1}{\cos\alpha}$