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Trigonometric Ratios of Multiples of an Angle

For all permi...

Question

For all permissible values of $A, 2 A$ , following holds true.
$(i) cot A + tan A = \frac{1}{sin A cos A} = 2 cosec 2 A (i i) cot A - tan A = \frac{{cos}^{2} A - {sin}^{2} A}{sin A cos A} = 2 cot 2 A (i i i) 2 cot A = 2 (cosec 2 A + cot 2 A) \Rightarrow cosec 2 A + cot 2 A = cot A$

Also to evaluate a series of form $f (x) + f (2 x) + f (4 x) + \dots + f (2^{n} x)$ when $f (x)$ can be expressed as $g (x) - g (2 x)$ , we can use the following technique,
$f (x) + f (2 x) + f (4 x) + \dots + f (2^{n} x) = (g (x) - g (2 x)) + (g (2 x) - g (4 x)) + \dots (g (2^{n} x) - g (2^{n + 1} x)) = g (x) - g (2^{n + 1} x)$

Based on the above information, solve the following questions for all permissible values of $x$ .

The value of $cosec 2 x + cosec 4 x + cosec 8 x + cosec 16 x + cosec 32 x$

cot x - cot 64 x

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cot 2 x - cot 64 x

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cot x - cot 32 x

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cot 2 x - cot 32 x

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Solution

The correct option is C $cot x - cot 32 x$
$∵ cosec 2 x + cot 2 x = cot x \Rightarrow cosec 2 x = cot x - cot 2 x$
Now,
$cosec 4 x = cot 2 x - cot 4 x cosec 8 x = cot 4 x - cot 8 x cosec 16 x = cot 8 x - cot 16 x cosec 32 x = cot 16 x - cot 32 x$
So,
$cosec 2 x + cosec 4 x + cosec 8 x + cosec 16 x + cosec 32 x = cot x - cot 32 x$

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