MCQ
$\sec {50^o} + \tan {50^o}$ is equal to
- A$\tan {20^o} + \tan {50^o}$
- B$2\tan {20^o} + \tan {50^o}$
- ✓$\tan {20^o} + 2\tan {50^o}$
- D$2\tan {20^o} + 2\tan {50^o}$
==> $\tan ({70^o} - {20^o}) = \frac{{\tan {{70}^o} - \tan {{20}^o}}}{{1 + \tan {{70}^o}\tan {{20}^o}}}$
==> $\tan {50^o} + \tan {70^o}\tan {20^o}\tan {50^o} = \tan {70^o} - \tan {20^o}$
==> $\tan {50^o} + \tan {50^o} = \tan {70^o} - \tan {20^o}$
$[\,\because \tan {70^o} = \cot {20^o}]$
==> $2\tan {50^o} + \tan {20^o} = \tan {70^o}$
==> $2\tan {50^o} + \tan {20^o} = \tan {50^o} + \sec {50^o}$.
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$f(0)=1 \text { and } \int_0^{\frac{\pi}{3}} f( t ) dt =0$
Then which of the following statements is (are) $TRUE$?
$(A)$ The equation $f( x )-3 \cos 3 x =0$ has at least one solution in $\left(0, \frac{\pi}{3}\right)$
$(B)$ The equation $f( x )-3 \sin 3 x =-\frac{6}{\pi}$ has at least one solution in $\left(0, \frac{\pi}{3}\right)$
$(C)$ $\lim _{x \rightarrow 0} \frac{x \int_0^x f(t) d t}{1- e ^{x^2}}=-1$
$(D)$ $\lim _{ x \rightarrow 0} \frac{\sin x \int_0^{ x } f( t ) dt }{ x ^2}=-1$