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Arithmetic Mean

Identity tran...

Question

Identity transformations of Trigonometric Expressions.
prove the following identities.
$\frac{2 s i n^{2} (\frac{π}{4} - α)}{c o s 2 α} = c o t (\frac{π}{4} + α) .$

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Solution

$\frac{2 s i n^{2} (\frac{π}{4} - α)}{c o s 2 α} = L H S$
$= \frac{1 - c o s (\frac{π}{2} - 2 α)}{c o s 2 α}$
$= \frac{1 - s i n 2 α}{c o s 2 α}$
$= \frac{(c o s α - s i n α^{2})}{c o s^{2} α - s i n^{2} α}$
$= \frac{c o s α - s i n α}{c o s α + s i n α}$
$= \frac{c o t α - 1}{c o s α + 1}$
$∵ (c o t \frac{π}{4} = 1)$
$= \frac{c o t α . c o t \frac{π}{4} - 1}{c o t α + c o t \frac{π}{4}}$
$= c o t (\frac{π}{4} + α)$
$= R H S$
Hence proved!

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