if (1 + tan 1°)(1 + tan 2°)(1 + tan 3°)...... ....)(1 + tan 44°)(1 + tan 45°) ^ = { ((((√(50)) + 7)))^(1/3) − ((((√(50))−7)))^(1/3) }^((x − 7)) find x = ...?


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Question Number 1999 by Fitrah last updated on 29/Oct/15
$if (1 + tan 1°)(1 + tan 2°)(1 + tan 3°)...... ....)(1 + tan 44°)(1 + tan 45°) ^ = { ((((√(50)) + 7)))^(1/3) − ((((√(50))−7)))^(1/3) }^((x − 7)) find x = ...?$
$i f$ $(1 + t a n 1 °) (1 + t a n 2 °) (1 + t a n 3 °) . . . . . .$ $Missing \left or extra \right$ $f i n d x = . . . ?$
	Answered by Yozzi last updated on 29/Oct/15

	$T h e g i v e n e q u a t i o n c a n b e w r i t t e n$ $a s d = c^{x - 7} w h e r e c, d > 0 s i n c e$ $t a n (r^{°}) > 0 \forall r \in [1, 45] \Rightarrow 1 + t a n (r^{°}) > 0$ $a n d \sqrt{50} + 7 > \sqrt{50} - 7 \Rightarrow {(\sqrt{50} + 7)}^{1 / 3} > {(\sqrt{50} - 7)}^{1 / 3}$ $\Rightarrow {(\sqrt{50} + 7)}^{1 / 3} - {(\sqrt{50} - 7)}^{1 / 3} > 0 .$ $∴ l n d = {l n c}^{x - 7} \Rightarrow l n d = (x - 7) l n c$ $\Rightarrow x - 7 = \frac{l n d}{l n c} = {l o g}_{c} d$ $x = 7 + {l o g}_{c} d$ $x = 7 + {l o g}_{({(\sqrt{50} + 7)}^{1 / 3} - {(\sqrt{50} - 7)}^{1 / 3})} (\underset{r = 1}{\prod^{45}} (1 + t a n (r^{°})))$
	Commented by Rasheed Soomro last updated on 29/Oct/15

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