Question

If $0 < \theta, \phi < \frac{\pi}{2}, x = \sum_{n=0}^{\infty} \cos^{2n}\theta, y = \sum_{n=0}^{\infty} \sin^{2n}\phi$ and $z = \sum_{n=0}^{\infty} \cos^{2n}\theta \sin^{2n}\phi$ then:

• A. xy - z = (x + y)z
• B. xy + yz + zx = z
• C. xyz = 4
• D. xy + z = (x + y)z

Answer 1

Answer: (D)

xy + z = (x + y)z

Answer 2

The problem requires evaluating three infinite geometric series and then finding the correct relationship between the resulting variables $x$, $y$, and $z$.

1. Evaluating the Series: The sum of an infinite geometric series is given by the formula $S = \frac{a}{1-r}$, where $a$ is the first term and $r$ is the common ratio, provided that $|r| < 1$.

• For $x = \sum_{n=0}^{\infty} \cos^{2n}\theta$: The first term ($a$) is $\cos^{2 \times 0}\theta = \cos^0\theta = 1$. The common ratio ($r$) is $\cos^2\theta$. Given $0 < \theta < \frac{\pi}{2}$, we know that $0 < \cos\theta < 1$, which implies $0 < \cos^2\theta < 1$. Thus, $|r| < 1$, and the series converges. Therefore, $x = \frac{1}{1 - \cos^2\theta}$. Using the fundamental trigonometric identity $\sin^2\theta + \cos^2\theta = 1$, we have $1 - \cos^2\theta = \sin^2\theta$. So, $x = \frac{1}{\sin^2\theta}$.

• For $y = \sum_{n=0}^{\infty} \sin^{2n}\phi$: The first term ($a$) is $\sin^{2 \times 0}\phi = \sin^0\phi = 1$. The common ratio ($r$) is $\sin^2\phi$. Given $0 < \phi < \frac{\pi}{2}$, we know that $0 < \sin\phi < 1$, which implies $0 < \sin^2\phi < 1$. Thus, $|r| < 1$, and the series converges. Therefore, $y = \frac{1}{1 - \sin^2\phi}$. Using the identity $\sin^2\phi + \cos^2\phi = 1$, we have $1 - \sin^2\phi = \cos^2\phi$. So, $y = \frac{1}{\cos^2\phi}$.

• For $z = \sum_{n=0}^{\infty} \cos^{2n}\theta \sin^{2n}\phi$: This series can be rewritten as $z = \sum_{n=0}^{\infty} (\cos^2\theta \sin^2\phi)^n$. The first term ($a$) is $(\cos^2\theta \sin^2\phi)^0 = 1$. The common ratio ($r$) is $\cos^2\theta \sin^2\phi$. Since $0 < \cos^2\theta < 1$ and $0 < \sin^2\phi < 1$, their product $0 < \cos^2\theta \sin^2\phi < 1$. Thus, $|r| < 1$, and the series converges. Therefore, $z = \frac{1}{1 - \cos^2\theta \sin^2\phi}$.

2. Expressing Trigonometric Terms in terms of x and y: From the expressions for $x$ and $y$:

• $x = \frac{1}{\sin^2\theta} \implies \sin^2\theta = \frac{1}{x}$. Using $\cos^2\theta = 1 - \sin^2\theta$, we get $\cos^2\theta = 1 - \frac{1}{x} = \frac{x-1}{x}$.

• $y = \frac{1}{\cos^2\phi} \implies \cos^2\phi = \frac{1}{y}$. Using $\sin^2\phi = 1 - \cos^2\phi$, we get $\sin^2\phi = 1 - \frac{1}{y} = \frac{y-1}{y}$.

3. Substituting into the expression for z: Now, substitute the expressions for $\cos^2\theta$ and $\sin^2\phi$ into the formula for $z$: $z = \frac{1}{1 - \cos^2\theta \sin^2\phi} = \frac{1}{1 - \left(\frac{x-1}{x}\right)\left(\frac{y-1}{y}\right)}$ To simplify the denominator: $1 - \frac{(x-1)(y-1)}{xy} = \frac{xy - (x-1)(y-1)}{xy}$ $= \frac{xy - (xy - x - y + 1)}{xy}$ $= \frac{xy - xy + x + y - 1}{xy}$ $= \frac{x + y - 1}{xy}$

So, $z = \frac{1}{\frac{x + y - 1}{xy}} = \frac{xy}{x + y - 1}$.

4. Rearranging the Relationship: The derived relationship is $z = \frac{xy}{x + y - 1}$. Multiply both sides by $(x + y - 1)$: $z(x + y - 1) = xy$ $zx + zy - z = xy$

5. Checking the Options: We need to find which of the given options is equivalent to $xy = z(x+y-1)$.

• (A) $xy - z = (x + y)z$ $xy - z = xz + yz$ $xy = xz + yz + z$ $xy = z(x+y+1)$ This does not match our derived equation.

• (B) $xy + yz + zx = z$ $xy = z - yz - zx$ $xy = z(1 - y - x)$ This does not match our derived equation.

• (C) $xyz = 4$ This option suggests a constant numerical relationship. However, the values of $x$, $y$, and $z$ depend on $\theta$ and $\phi$. For example, if $\theta = \pi/4$ and $\phi = \pi/4$: $\cos^2(\pi/4) = (1/\sqrt{2})^2 = 1/2$. $\sin^2(\pi/4) = (1/\sqrt{2})^2 = 1/2$. $x = \sum_{n=0}^{\infty} (1/2)^n = \frac{1}{1 - 1/2} = 2$. $y = \sum_{n=0}^{\infty} (1/2)^n = \frac{1}{1 - 1/2} = 2$. $z = \sum_{n=0}^{\infty} (1/2 \times 1/2)^n = \sum_{n=0}^{\infty} (1/4)^n = \frac{1}{1 - 1/4} = \frac{1}{3/4} = 4/3$. In this case, $xyz = 2 \times 2 \times 4/3 = 16/3 \neq 4$. So, this option is incorrect.

• (D) $xy + z = (x + y)z$ $xy + z = xz + yz$ $xy = xz + yz - z$ $xy = z(x+y-1)$ This exactly matches our derived relationship.

Therefore, the correct option is (D).