For 0- - 2- if x- n-0 -cos 2n - -y- n-0 -sin 2n - and z- n-0 -cos 2n -sin 2n - then show that xyz-xy-z-

We have #160;x=∑ _ n=0 ^∞ #160;cos ^ 2n ϕ #160; #160; = 1 / 1 cos ^ 2 ϕ #160; #160; = ^ 2 ϕ #160; #160;y=∑ _ n=0 ^∞ #160;sin ^ 2n ϕ #1 ...

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Byju's Answer

Standard XII

Mathematics

Polar Representation of a Complex Number

For 0<ϕ <π/...

Question

For $0 < ϕ < \frac{π}{2}$ , if $x = \sum_{n = 0}^{\infty} {cos}^{2 n} ϕ, y = \sum_{n = 0}^{\infty} {sin}^{2 n} ϕ$ and $z = \sum_{n = 0}^{\infty} {cos}^{2 n} ϕ {sin}^{2 n} ϕ$ , then show that $x y z = x y + z$ .

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Solution

We have $x = \infty \sum n = 0 {cos}^{2 n} ϕ = \frac{1}{1 - {cos}^{2} ϕ} = {csc}^{2} ϕ$
$y = \infty \sum n = 0 {sin}^{2 n} ϕ = \frac{1}{1 - {sin}^{2} ϕ} = {sec}^{2} ϕ$
$z = \infty \sum n = 0 {cos}^{2 n} ϕ {sin}^{2 n} ϕ = \frac{1}{1 - {sin}^{2} ϕ {cos}^{2} ϕ}$
$= \frac{{sin}^{2} ϕ + {cos}^{2} ϕ}{{sin}^{2} ϕ {cos}^{2} ϕ} = \frac{1}{{cos}^{2} ϕ} + \frac{1}{{sin}^{2} ϕ} = x + y$
$\Rightarrow x y z - z = x y \Rightarrow x y z = x y + z$

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