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Geometry Level 1

If $\theta$ is an acute angle and $\tan\theta+\cot\theta=2$ , find the value of $\tan^7\theta+\cot^7\theta$ .

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2 solutions

Rishabh Jain
Jun 20, 2016

Relevant wiki: Trigonometric Equations - Problem Solving - Easy

$\tan\theta+\dfrac1{\tan\theta}=2~~~\left(\small{\because\cot\theta=\frac1{\tan\theta}}\right)$

$\implies (\tan\theta-1)^2=0\implies \tan\theta=1$

$\tan^7\theta+\cot^7\theta=1+1=\large\color{#3D99F6}{\boxed 2}$

Akshat Sharda
Jun 20, 2016

$\tan \theta+\cot \theta=2 \Rightarrow \frac{\sin^2 \theta+\cos^2 \theta}{\sin \theta \cos \theta}=2 \\ 1=2 \sin \theta \cos \theta \Rightarrow 1=\sin 2 \theta \Rightarrow \theta =\frac{\pi}{4} \\ \Rightarrow \tan^7 \frac{\pi}{4}+\cot ^7 \frac{\pi}{4}=\boxed{2}$

Small note: $\theta = \dfrac{\pi}{4}+k \pi$ where $k$ is an integer. However, since $\tan$ and $\cot$ have period $\pi$ , this doesn't raise any issues.

Michael Fuller - 4 years, 11 months ago

Can you explain in a bit more?

vivek kushal - 4 years, 10 months ago

Sure, the $\tan$ and $\cot$ graphs have period $\pi$ , which means that $\tan(\theta) \equiv \tan(\theta+k \pi)$ where $k$ is an integer. In other words, the $\tan$ graph repeats itself after every $180^{\circ}$ .

Similarly, the $\sin$ graph repeats itself every $2\pi$ , so $1=\sin 2 \theta \Rightarrow 2 \theta = \dfrac{\pi}{2} \color{#D61F06}{+ 2k \pi} \Rightarrow \theta = \dfrac{\pi}{4} + k \pi$ as I mentioned.

I always find these types of questions easier when visualising a sketch. Think about the line $y=1$ and $y=\sin x$ . An obvious solution is $\dfrac{\pi}{2}$ , but there's another solution every $\pm 2k\pi$ .

Michael Fuller - 4 years, 10 months ago

@Michael Fuller – Got it...thanks Michael :)

vivek kushal - 4 years, 10 months ago

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