If $sin\theta +cos\theta=\sqrt{3}$, then prove that $tan\theta+cot\theta=1$.

Given: $sin\theta +cos\theta=\sqrt{3}$.

To do: To prove that $tan\theta+cot\theta=1$.

Solution: 

As given, $sin\theta +cos\theta=\sqrt{3}$

On Squaring both sides,

$\Rightarrow ( sin\theta+cos\theta)=( \sqrt{3})^{2}$

$\Rightarrow sin^{2}\theta+cos^{2}\theta+2sin\theta.cos\theta=3$

$\Rightarrow 1+2sin\theta cos\theta=3$

$\Rightarrow 2sin\theta cos\theta=3-1$

$\Rightarrow 2sin\theta cos\theta=2$

$\Rightarrow sin\theta cos\theta=1$ .......... $( 1)$

Now, $tan\theta+cot\theta$

$=\frac{sin\theta}{cos\theta}+\frac{cos\theta}{sin\theta}$

$=\frac{sin^{2}\theta+cos^{2}\theta}{sin\theta cos\theta}$

$=\frac{1}{1}=1$            [$\because sin^{2}\theta+cos^{2}\theta=1\ and\ sin\theta cos\theta=1,\ from\ ( 1)$]

Hence, $tan\theta+cot\theta=1$

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Updated on: 10-Oct-2022

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