If tanθ = \frac{1}{\sqrt{7}} then \frac{\left(\right. \left(cosec\right)^{2} \theta - \left(sec\right)^{2} \theta \left.\right)}{\left(\right. \left(cosec\right)^{2} \theta + \left(sec\right)^{2} \theta \left.\right)} = ?

Foundation

Mathematics Foundation

Introduction to Trigonometry

Question

If $tanθ = \frac{1}{\sqrt{7}}$ then $\frac{({cosec}^{2} θ - \sec^{2} θ)}{({cosec}^{2} θ + \sec^{2} θ)} = ?$

$\frac{- 3}{4}$

$\frac{2}{3}$

$\frac{3}{4}$

$\frac{- 2}{3}$

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Given tanθ = 1/√7, we can construct a right triangle where opposite side = 1, adjacent side = √7. The hypotenuse is √(1² + (√7)²) = √8 = 2√2.

Therefore, cosecθ = hypotenuse/opposite = 2√2/1 = 2√2 and secθ = hypotenuse/adjacent = 2√2/√7.

We need to find (cosec²θ - sec²θ)/(cosec²θ + sec²θ).

Substitute the values: cosec²θ = (2√2)² = 8 and sec²θ = (2√2/√7)² = 8/7.

So, numerator = 8 - 8/7 = (56 - 8)/7 = 48/7.

Denominator = 8 + 8/7 = (56 + 8)/7 = 64/7.

The expression becomes (48/7) / (64/7) = 48/64 = 3/4.

Final answer: $\frac{3}{4}$