Question

Prove That
(tan20/cosec70)^2 + (cot20/sec70)^2 + 2 tan15tan37tan53tan60tan75 = 1

Answer 1

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Answer 2

$\Large{\underline{\underline{\tt{\purple{Hello !!}}}}}$

$= ( \frac{tan20}{cosec70} ) {}^{2} + ( \frac{cot20}{sec70} ) {}^{2} + 2tan15.tan60.tan \: 75$

$= ( \frac{ \frac{sin20}{cos \: 20} }{ \frac{1}{sin 70} } ) {}^{2} + ( \frac{ \frac{cos20}{sin20} }{ \frac{1}{cos \: 70} } ) {}^{2} + 2 \: tan \: 15.tan \: 60. \: tan \: (90 - 15)$

$= ( \frac{sin \: 20}{cos \: 20 } \times sin 70) \: {}^{2} \: + ( \frac{cos \: 20}{sin \: 20} \times cos \: 70) + 2 \: tan \: 15.tan \: 60.cot \: 15$

$= ( \frac{sin20}{cos \: 20} \times cos \: 20) {}^{2} + ( \frac{cos20}{sin20} \times sin20) {}^{2} + 2.tan.60(1)$

$= sin {}^{2} 20 + cos \: {}^{2} 20 + 2 \: tan \: 60$

$= 1 + 2( \sqrt{3} )$

$= 1 + 2 \sqrt{3}$

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