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Arithmetic Mean

Identity tran...

Question

Identity transformations of Trigonometric Expressions.
prove the following identities.
$1 - s i n 4 α + c o t (\frac{3}{4} π - 2 α) c o s 4 α = 0.$

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Solution

$c o t (\frac{3 π}{4} - 2 α) = \frac{s i n 4 α - 1}{c o s 4 α}$
LHS $= \frac{c o t \frac{3 π}{4} . c o t 2 α - 1}{c o t \frac{3 π}{4} + c o t 2 α}$
$= \frac{1 - c o t 2 α}{1 + c o t 2 α} = \frac{s i n 2 α - c o s 2 α}{s i n 2 α + c o s 2 α} \times (\frac{s i n 2 α - c o s 2 α}{s i n 2 α - c o s 2 α})$
$= \frac{(s i n 2 α - c o s 2 α)^{2}}{- c o s 4 α} = \frac{s i n^{2} 2 α + c o s^{2} 2 α - 2 c o s 2 α s i n 2 α}{- c o s 4 α}$
$= \frac{1 - s i n 4 α}{- c o s 4 α} = \frac{s i n 4 α - 1}{c o s 4 α} = R H S$
Hence proved!

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