Question

For x, y, z $ϵ(0,\frac{\pi }{2}),$ let x, y, z be first three consecutive terms of an arithmetic progression such that cos x + cox y + cos z = 1 and sin x + sin y + sin z = $\frac{1}{\sqrt{2}}$, then which of the following is/are correct?

• A. $coty=\sqrt{2}$
• B. $cos(x-y)=\frac{\sqrt{3}-1}{2\sqrt{2}}$
• C. $tan2y=\frac{2\sqrt{2}}{3}$
• D. $sin(x-y)+sin(z-y)=0$

Answer 1

Answer: (A), (D)

The correct options are
A

$coty=\sqrt{2}$

D

$sin(x-y)+sin(z-y)=0$

We have y - d = x, y, y + d = z in AP

Now $\sum$ cos x = 1 $\Rightarrow$ cos (y - d) + cos y + cos (y + d) = 1 $\Rightarrow$ cos y(2 cos d + 1) = 1 .....(1)

Also, $\sum sinx=\frac{1}{\sqrt{2}}\Rightarrow sin(y-d)+siny+sin(y+d)=\frac{1}{\sqrt{2}}\Rightarrow siny(2cosd+1)=\frac{1}{\sqrt{2}}$ .....(2)

$\therefore \frac{Equation\left(1\right)}{Equation\left(2\right)}\Rightarrow coty=\sqrt{2}$

Now, putting cos $y=\frac{\sqrt{2}}{\sqrt{3}}$ in (1), we get

2 cos d + 1 = $\frac{\sqrt{2}}{\sqrt{3}}\Rightarrow cosd=\frac{\sqrt{3}-\sqrt{2}}{2\sqrt{2}}=cos(y-x)=cos(x-y)$

Also, tan 2y = $\frac{2tany}{1-ta{n}^{2}y}=\frac{2\times \frac{1}{\sqrt{2}}}{1-\frac{1}{2}}=\frac{\sqrt{2}}{\frac{1}{2}}=2\sqrt{2}$

Clearty, sin (x-y) + sin(z-y) = sin(-d) + sin d = 0.