Track / Stage 5 / 9 of 400 #609 of 1964

Problem 609

AIME late

Algebra Difficulty 5.0 Find the answer

3. Let $a \neq \frac{k \pi}{2}(k=0, \pm 1, \pm 2, \cdots), T \equiv \frac{\sin a+\tan a}{\cos a+\cot a}$ .

Pick one

Official solution

(C)
3. 【Analysis and Solution】By making an identity transformation, we easily get
$T=\frac{\sin a+\tan a}{\cos a+\cot a}=\frac{\tan a(\cos a+1)}{\cot a(\sin a+1)}=\frac{\left(\tan ^{2} a\right)(\cos a+1)}{\sin a+1}>0 \text {. }$
(Where $\left.a \neq \frac{k \pi}{2}, k=0, \pm 1, \pm 2, \cdots\right)$ ).

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.