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If $tan\theta+cot\theta=5$, then find the value of $tan^{2}\theta+cot^{2}\theta$.
Given: $tan\theta +cot\theta =5$.
To do: To find the value of $tan^{2}\theta +cot^{2}\theta$.
Solution:
As given, $tan\theta +cot\theta =5$
Now squaring both sides we get,
$\Rightarrow (tan^{2}\theta +cot^{2}\theta +2.tan\theta .cot\theta )=25$
$\Rightarrow tan^{2}\theta +cot^{2}\theta =25−2$ [ Since $tan\theta .cot\theta =1$]
$\Rightarrow tan^{2}\theta +cot^{2}\theta =23$
Thus, $tan^{2}\theta +cot^{2}\theta =23$.
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