If x is an acute angle and tan x = 1/√7 , then the value of (cosec² x − sec² x)/(cosec² x + sec² x) is(a) 3/4(b) 1/2(c) 2(d) 5/4

Question

\[ \text{If } x \text{ is an acute angle and } \tan x=\frac{1}{\sqrt7}, \]

\[ \text{then the value of } \frac{\cosec^2x-\sec^2x}{\cosec^2x+\sec^2x} \]

(a) \(\frac34\)
(b) \(\frac12\)
(c) \(2\)
(d) \(\frac54\)

\[ \tan x=\frac{1}{\sqrt7} \]

Take

\[ \text{Perpendicular}=1,\quad \text{Base}=\sqrt7 \]

\[ \text{Hypotenuse} = \sqrt{1+7} = 2\sqrt2 \]

\[ \sin x=\frac{1}{2\sqrt2} \Rightarrow \cosec^2x=8 \]

\[ \cos x=\frac{\sqrt7}{2\sqrt2} \Rightarrow \sec^2x=\frac87 \]

Now,

\[ \frac{\cosec^2x-\sec^2x}{\cosec^2x+\sec^2x} = \frac{8-\frac87}{8+\frac87} \]

\[ = \frac{\frac{56-8}{7}}{\frac{56+8}{7}} = \frac{48}{64} \]

\[ =\frac34 \]

\[ \boxed{\frac34} \]

Correct Option: (a)

Spread the love