Question

If Theeta is an acute angle and Tan Theeta + Cot Theeta = 2 Find the value of tan ^7 Theeta + Cot^7 Theeta.

Answer 1

$\textbf{ solution–}$

We have

$tanθ + cot θ = 2\\ \\ tanθ + \frac{1}{cotθ} = 2 \\ \\ \frac{ {tan}^{2}θ + 1}{tanθ} = 2 \\ \\ {tan}^{2} θ + 1 = 2tanθ \\ \\ {tan}^{2} θ + 1 - 2tanθ = 0$
From

(a – b)² = a² + b² – 2ab

$(tanθ - 1)^{2} = 0 \\ \\ tanθ - 1 = 0 \\ \\ tanθ = 1 \\ tanθ = tan45° \\ θ = 45°$
Now,

${tan}^{7} θ + {cot}^{7} θ$

Putting θ =45°

$= {tan}^{7} 45° + {cot}^{7} 45° \\ = (tan45°) ^{7} + (cot45°) ^{7} \\ = {1}^{7} + {1}^{7} \\ = 2$

$\textbf{ So, the value is 2}$

Answer 2

Answer:

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