First slide

Trigonometric Identities

Question

For $0 < ϕ < π / 2, if x = \sum_{n = 0}^{\infty} \cos^{2 n} ϕ, y = \sum_{n = 0}^{\infty} \sin^{2 n} ϕ$ , $z = \sum_{n = 0}^{\infty} \cos^{2 n} {ϕsin}^{2 n} ϕ,$ then

Moderate

A

xyz=xz+y
B

xyz=xy+z
C

xyz=x+y+z
D

xyz=yz+x

Solution

For GP sum infinite term is a/1-r
All are infinite geometric progression with common ratio < I
$\begin{array}{l} x = \frac{1}{1 - \cos^{2} ϕ} = \frac{1}{\sin^{2} ϕ}, y = \frac{1}{1 - \sin^{2} ϕ} = \frac{1}{\cos^{2} ϕ}, z = \frac{1}{1 - \cos^{2} {ϕsin}^{2} ϕ} \\ now, xy + z = \frac{1}{\sin^{2} {ϕcos}^{2} ϕ} + \frac{1}{1 - \sin^{2} {ϕcos}^{2} ϕ} \end{array}$
$\begin{array}{l} = \frac{1}{\sin^{2} {ϕcos}^{2} ϕ (1 - \sin^{2} {ϕcos}^{2} ϕ)} \\ xy + z = xyz - - - (i) \end{array}$
clearly, $x + y = \frac{\sin^{2} ϕ + \cos^{2} ϕ}{\sin^{2} ϕcos ϕ} = xy$
$∴ x + y + z = xyz$ [using Eq. (i)]

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