If tan⁸ θ + cot⁸ θ = m, then what is the value of tan θ + cot θ ? 

Formula used:

(x + \(\frac{1}{x}\))² = (x² + \(\frac{1}{x^2}\)) + 2

(x + \(\frac{1}{x}\)) = \(\sqrt{[(x^2 + \frac{1}{x^2}) + 2]}\)

Calculation:

Given that 

tan8 θ + cot8 θ = m

Since, cot θ = 1/tan θ. Therefore,

tan⁸θ + (1/tan²θ) = m

Let tan θ = n then 

n⁸ + 1/n⁸ = m      ---(1)

We know that , (x + \(\frac{1}{x}\)) = \(\sqrt{[(x^2 + \frac{1}{x^2}) + 2]}\)

⇒ (n⁴ + 1/n⁴) = \(\sqrt{[(n^8 + \frac{1}{n^8}) + 2]}\)

⇒ (n4 + 1/n4) = \(\sqrt{[m + 2]}\)

Again applying the same identity

⇒ (n² + 1/n²) = \(\sqrt{(\sqrt{(m + 2)}+2)}\)

Again applying the same identity

⇒ (n + 1/n) = \(\sqrt {\sqrt{(\sqrt{(m + 2)}+2)}+2}\)

But, n = tan θ 

⇒ tan θ + 1/tan θ = \(\sqrt {\sqrt{(\sqrt{(m + 2)}+2)}+2}\)

∴ tan θ + cot θ = \(\sqrt{\sqrt{\sqrt{m+2}+2}+2}\)