If $0 < θ, ? < \frac{π}{2}, x = \sum_{n = 0}^{\infty} c o s^{2 n} θ, y = \sum_{}^{} s i n^{2 n} ? a n d z = \sum_{n = 0}^{\infty} c o s^{2 n} θ . s i n^{2 n} ? t h e n :$

Option 1 -

xy + yz + zx = z

Option 2 -

xy – z = (x + y) z

Option 3 -

xyz = 4

Option 4 -

xy + z = (x + y) z

2 Views | Posted a month ago

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1 Answer

Answered by

alok kumar singh | Contributor-Level 10

a month ago

Correct Option - 2

Detailed Solution:

$x = \sum_{n = 0}^{\infty} c o s^{2 n} θ = c o s^{0} θ + c o s^{2} θ + c o s^{4} θ + . . . . . = 1 + c o s^{2} θ + c o s^{4} θ + . . . . .$

a = 1, r = cos² θ

$x = S_{\infty} = \frac{a}{1 - r} = \frac{1}{1 - c o s^{2} θ} = \frac{1}{s i n^{2} θ}$

Similarly, y = $\frac{1}{c o s^{2} θ} \Rightarrow \frac{1}{y} = c o s^{2} θ$

$z = \frac{x y}{x y - 1} \Rightarrow x y z - z = x y . . . . . . . . . . . (i)$

Also, $\frac{1}{x} + \frac{1}{y} = 1 \Rightarrow x + y = x y . . . . . . . . . . (i i)$

(i) & (ii) ->xyz = xy + z -> (x + y) z = xy + z

<math> <mrow> <mi>x</mi> <mo>=</mo> <mstyle displaystyle="true"> <munderover> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <mo>∞</mo> </mrow> </munderover> <mrow> <mi>c</mi> <mi>o</mi> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mi>θ</mi> <mo>=</mo> <mi>c</mi> <mi>o</mi> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msup> <mi>θ</mi> <mo>+</mo> <mi>c</mi> <mi>o</mi> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mi>θ</mi> <mo>+</mo> <mi>c</mi> <mi>o</mi> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msup> <mi>θ</mi> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>c</mi> <mi>o</mi> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mi>θ</mi> <mo>+</mo> <mi>c</mi> <mi>o</mi> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msup> <mi>θ</mi> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </mstyle> </mrow> </math>  a = 1, r = cos2 θ <math> <mrow> <mi>x</mi> <mo>=</mo> <msub> <mrow> <mi>S</mi> </mrow> <mrow> <mo>∞</mo> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>r</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>c</mi> <mi>o</mi> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mi>θ</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>s</mi> <mi>i</mi> <msup> <mrow> <mi>n</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mi>θ</mi> </mrow> </mfrac> </mrow> </math>           Similarly, y = <math> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>c</mi> <mi>o</mi> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mi>θ</mi> </mrow> </mfrac> <mo>⇒</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>y</mi> </mrow> </mfrac> <mo>=</mo> <mi>c</mi> <mi>o</mi> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mi>θ</mi> </mrow> </math>   <math> <mrow> <mi>z</mi> <mo>=</mo> <mfrac> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mrow> <mi>x</mi> <mi>y</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mo>⇒</mo> <mi>x</mi> <mi>y</mi> <mi>z</mi> <mo>−</mo> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mi>y</mi> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math>           Also, <math> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>x</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>y</mi> </mrow> </mfrac> <mo>=</mo> <mn>1</mn> <mo>⇒</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mi>y</mi> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mi>i</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math>   (i) & (ii) ->xyz = xy + z -> (x + y) z = xy + z

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If $0 < θ, ? < \frac{π}{2}, x = \sum_{n = 0}^{\infty} c o s^{2 n} θ, y = \sum_{}^{} s i n^{2 n} ? a n d z = \sum_{n = 0}^{\infty} c o s^{2 n} θ . s i n^{2 n} ? t h e n :$

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If 0 < θ , ? < π 2 , x = ∑ n = 0 ∞ c o s 2 n θ , y = ∑ s i n 2 n ? a n d z = ∑ n = 0 ∞ c o s 2 n θ . s i n 2 n ? t h e n :

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If $0 < θ, ? < \frac{π}{2}, x = \sum_{n = 0}^{\infty} c o s^{2 n} θ, y = \sum_{}^{} s i n^{2 n} ? a n d z = \sum_{n = 0}^{\infty} c o s^{2 n} θ . s i n^{2 n} ? t h e n :$