Question

If sin theta +cos theta =√3 then prove tan theta +cot theta = 1

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Answer 1

ANSWER:-

Given:

If sin theta + cos theta= √3

To prove:

tan theta + cot theta= 1.

Proof:

$sin \theta \: + cos \theta = \sqrt{3} \\ squaring \:both \: sides \: we \: get \\ = > (sin \theta + cos \theta) {}^{2} = ( { \sqrt{3} )}^{2} \\ \\ = > {sin}^{2} \theta + cos {}^{2} \theta + 2sin \theta \: cos \theta = 3 \\ \\ = > 2sin \theta \: cos \theta = 3 - 1 = 2 \: \: \: \: \: ( {sin}^{2} \theta + {cos}^{2} \theta = 1) \\ \\ = > sin \theta \: cos \theta = 1.............(1) \\ \\ = > tan \theta + cot \theta \\ \\ = > \frac{sin \theta}{cos \theta} + \frac{cos \theta}{sin \theta} \\ \\ = > \frac{ {sin}^{2} \theta + {cos}^{2} \theta}{cos \theta \: sin \theta} = \frac{1}{1} \: \: \: \: \: \: [using \: eq.(1)] \\ \\ = > 1 \\ \\ = > tan \theta + cot \theta = 1$

Proved.

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Answer 2

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