For all permissible values of A,2A, following holds true. (i)cotA+tanA=1sinAcosA=2cosec 2A(ii)cotA−tanA=cos2A−sin2AsinAcosA=2cot2A(iii)2cotA=2(cosec 2A+cot2A) ⇒cosec 2A+cot2A=cotA Also to evaluate a series of form f(x)+f(2x)+f(4x)+⋯+f(2nx) when f(x) can be expressed as g(x)−g(2x), we can use the following technique, f(x)+f(2x)+f(4x)+⋯+f(2nx)=(g(x)−g(2x))+(g(2x)−g(4x))+⋯(g(2nx)−g(2n+1x))=g(x)−g(2n+1x) Based on the above information, solve the following questions for all permissible values of x. The value of cosec 2x+cosec 4x+cosec 8x+cosec 16x+cosec 32x

For all permissible values of A,2A, following holds true. (i)cotA+tanA

1.	For all permissible values of A,2A, following holds true. (i)cotA+tanA=1sinAcosA=2cosec 2A(ii)cotA−tanA=cos2A−sin2AsinAcosA=2cot2A(iii)2cotA=2(cosec 2A+cot2A) ⇒cosec 2A+cot2A=cotA Also to evaluate a series of form f(x)+f(2x)+f(4x)+⋯+f(2nx) when f(x) can be expressed as g(x)−g(2x), we can use the following technique, f(x)+f(2x)+f(4x)+⋯+f(2nx)=(g(x)−g(2x))+(g(2x)−g(4x))+⋯(g(2nx)−g(2n+1x))=g(x)−g(2n+1x) Based on the above information, solve the following questions for all permissible values of x. The value of cosec 2x+cosec 4x+cosec 8x+cosec 16x+cosec 32x
Answer» 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 $cosec 2 x + cosec 4 x + cosec 8 x + cosec 16 x + cosec 32 x$