Question

If sin theeta+ cos theeta = √3 , then prove that tan 0 + cot 0 = 1.​

Answer 1

✓ Given :

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

✓ To find :

tan 0° + cot 0° = 1

✓ Solution :

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

Squaring on both the sides ,

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

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

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

$\therefore \:( { \sin }^{2} \theta \: + { \cos }^{2} \theta \: = 1)$

$\longrightarrow \: 2 \sin \theta \: \cos \theta \: = 2$

$= > \sin \theta \: \times \cos \theta = 1$

Since we have solution in sin and cos terms , Let's change over here too so that we can prove easily !

$\frac{ \sin( \:\theta) }{ \cos( \theta) } + \frac{ \cos( \theta) }{ \sin( \theta) } = 1$

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

$= \frac{1}{1} = 1$

$hence \: we \: proved \: that \: \\ \\ \tan \theta + \cot \theta = 1$

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