Question

If 'θ'  is an acute angle and cosec θ = \(\sqrt{2\sqrt{2\sqrt2}}..........\), then the value of tan θ 
1. 1
2. √2 
3. \(\frac{1}{\sqrt3}\)
4. √3

Answer 1

Correct Answer - Option 3 : \(\frac{1}{\sqrt3}\)

Concept:

cosec 30° = 2

tan 30° = \(\frac{1}{\sqrt3}\)

Calculation:

Given: cosec θ = \(\sqrt{2\sqrt{2\sqrt2}}..........\)

Let, x = \(\sqrt{2\sqrt{2\sqrt2}}..........\)

On squaring both sides we get, 

x² = \(2 \sqrt{2\sqrt2}..........\)

\(\rm ⇒ x^2=2x\)                    (∵ x = \(\sqrt{2\sqrt{2\sqrt2}}..........\))

\(\rm ⇒ x^2-2x=0\)

⇒ x(x - 2) = 0

⇒ x = 2 OR x = 0

If x = 2:

cosec θ = 2

⇒ θ = 30° 

∴ tan θ = tan 30° = \(\frac{1}{\sqrt3}\)

Hence, option (3) is correct.