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Trigonometric Identities — Maths STD 10 — Question

Maharashtra BoardEnglish MediumSTD 10MathsTrigonometric Identities2 Marks
Question
Very-Short and Short-Answer Questions.
Write the value of $\tan1^\circ\tan2^\circ\dots\tan89^\circ.$
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Answer

$\tan1^\circ\tan2^\circ\dots\tan89^\circ.$
$=\big[\tan1^\circ\tan89^\circ\big]\big[\tan2^\circ\tan88^\circ\big]\dots\big[\tan44^\circ\tan66^\circ\big]\tan45^\circ$
$=\big[\tan1^\circ\tan(90^\circ-1^\circ)\big]\\\ \ \big[\tan2^\circ\tan(90^\circ-2^\circ)\big]\dots\big[\tan40^\circ\tan(90^\circ-44^\circ)\big]\dots(1)$
$=\big[\tan1^\circ\cot1^\circ\big]\big[\tan2^\circ\cot2^\circ\big]\dots\big[\tan44^\circ\cot44^\circ\big]$
$=1\times1\times\dots\times1$
$=1$

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