Question

The value of tan160°(cot80° - tan80°) is​

Answer 1

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Answer 2

$\red{Value \: of \: tan 160\degree ( cot 80\degree - tan 80\degree ) }$

$= tan 160\degree\Big( \frac{cos 80\degree }{sin 80\degree} - \frac{sin 80\degree }{cos 80\degree }\Big)$

$= tan 160\degree \Big( \frac{cos^{2} 80 \degree - sin^{2} 80\degree }{ sin 80\degree cos 80 \degree }\Big)$

$= tan 160\degree \Big( 2(\frac{cos^{2} 80 \degree - sin^{2} 80\degree) }{ 2sin 80\degree cos 80 \degree }\Big)$

$= tan 160\degree \Big (\frac{ cos \: (2\times 80 \degree) }{sin \: (2 \times 80 \degree ) } \Big)$

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/* We know that , */

$\pink { 1. cos^{2} \theta - sin^{2} \theta = cos 2\theta }$

$\blue { 2. 2sin\theta cos \theta = sin 2\theta }$

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$= tan 160\degree \Big ( 2 \frac{cos \: 160\degree }{sin \:160\degree }\Big)$

$= 2 \times tan 160\degree cot 160\degree$

$\boxed { \pink { \because tan \theta cot \theta = 1 }}$

$= 2 \times 1$

$= 2$

Therefore.,

$\red{Value \: of \: tan 160\degree ( cot 80\degree - tan 80\degree ) }\green { = 1 }$

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