Prove the following


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

and $z=\sum_{n=0}^{\infty} \cos ^{2 n} \theta \cdot \sin ^{2 n} \phi$ then :

  1. $x y-z=(x+y) z$

  2. $x y+y z+z x=z$

  3. $x y z=4$

  4. $x y+z=(x+y) z$

Correct Option: , 4


$x=\frac{1}{1-\cos ^{2} \theta} \Rightarrow \sin ^{2} \theta=\frac{1}{x}$

Also, $\cos ^{2} \theta=\frac{1}{y} \& 1-\sin ^{2} \theta \cos ^{2} \theta=\frac{1}{z}$

So, $1-\frac{1}{x} \times \frac{1}{y}=\frac{1}{z} \Rightarrow z(x y-1)=x y$.....(1)

Also, $\frac{1}{x}+\frac{1}{y}=1 \quad \Rightarrow x+y=x y$.........(2)

From (i) and (ii)

$x y+z=x y z=(x+y) z$

Leave a comment


Click here to get exam-ready with eSaral

For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.

Download Now