Trigonometrical Ratios And Identities Ques 43
Let $\theta \in (0, \frac{\pi}{4})$ and $t _1=(\tan \theta)^{\tan \theta}, t _2=(\tan \theta)^{\cot \theta}$, $t _3=(\cot \theta)^{\tan \theta}$ and $t _4=(\cot \theta)^{\cot \theta}$, then
(a) $t_1 > t_2 > t_3 > t_4$
(b) $t_4 > t_3 > t_1> t_2$
(c) $t_3 > t_1 > t_2 > t_4$
(d) $t_2 > t_3 > t_1 > t_4$
(2006, 3M)
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Answer:
Correct Answer: 43.(b)
Solution:
Formula:
Domain and Range of Trigonometric Functions:
- As when $\theta \in (0, \frac{\pi}{4}), \tan \theta<\cot \theta$
Since, $\quad \tan \theta<1$ and $\cot \theta>1$
$\therefore \quad(\tan \theta)^{\cot \theta}<1$ and $(\cot \theta)^{\tan \theta}>1$
$\therefore t _4>t _1$ which only holds in (b).
Therefore, (b) is the answer.
BETA
