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