Question

tanθ + cotθ = 2 find the value of tan²⁰¹⁹θ + cot²⁰²⁰θ = _____
$\tanθ + \cotθ = 2 \\ find \: the \: value \: of \: \\ \tan ^{2019} θ + \cot ^{2020} θ = x$
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Answer 1

Answer:

Step-by-step explanation:

We have,

$\tt{tan(\theta)+cot(\theta)=2}$

$\sf{\implies\,\dfrac{sin(\theta)}{cos(\theta)}+\dfrac{cos(\theta)}{sin(\theta)}=2}$

$\sf{\implies\,\dfrac{sin^2(\theta)+cos^2(\theta)}{cos(\theta)\,sin(\theta)}=2}$

$\sf{\implies\,\dfrac{1}{cos(\theta)\,sin(\theta)}=2}$

$\sf{\implies\,2cos(\theta)\,sin(\theta)=1}$

$\sf{\implies\,sin(2\theta)=1}$

$\sf{\implies\,2\theta=\dfrac{\pi}{2}}$

$\sf{\implies\,\theta=\dfrac{\pi}{4}}$

Now,

$\sf{tan^{2019}(\theta)+cot^{2020}(\theta)}$

$\sf{=tan^{2019}\left(\dfrac{\pi}{4}\right)+cot^{2020}\left(\dfrac{\pi}{4}\right)}$

$\sf{=1+1}$

$\sf{=2}$