_ r = 1 ^ 100 , 2^ r - 1 , 2^r is equal to - Vidyadip

MCQ

$\sum\limits_{r = 1}^{100} {\frac{{\tan \,{2^{r - 1}}}}{{\cos \,{2^r}}}} $ is equal to

A
$tan\,2^{99} -tan\,1$
B
$tan\,2^{100}$
✓
$tan\,2^{100} -tan\,1$
D
none of these

✓

Answer

Correct option: C.

$tan\,2^{100} -tan\,1$

c
$\mathrm{T}_{\mathrm{x}}=\frac{\tan 2^{\mathrm{r}-1}}{\cos 2^{\mathrm{r}}}=\tan 2^{\mathrm{r}}-\tan 2^{\mathrm{r}-1}$

$ \Rightarrow \sum\limits_{r = 1}^{100} {{T_r} = \tan {2^{100}} - \tan 1} $

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