Question

show that (sin0+cos pi÷6)( sin pi÷6+tan pi÷6+tan pi÷4) ÷(tan pi÷6+cot pi÷3)(sec pi ÷3-cosec pi÷3)​

Answer 1

SOLUTION

TO EVALUATE

$\displaystyle \sf{ \frac{ (\sin 0 + \cos \frac{ \pi }{6})( \sin \frac{\pi}{6} + \tan \frac{\pi}{6} + \tan \frac{\pi}{4})}{(\tan \frac{\pi}{6} + \cot \frac{\pi}{3} )( \sec \frac{\pi}{3} - \cosec \frac{\pi}{3})} }$

EVALUATION

$\displaystyle \sf{ \frac{ (\sin 0 + \cos \frac{ \pi }{6})( \sin \frac{\pi}{6} + \tan \frac{\pi}{6} + \tan \frac{\pi}{4})}{(\tan \frac{\pi}{6} + \cot \frac{\pi}{3} )( \sec \frac{\pi}{3} - \cosec \frac{\pi}{3})} }$

$\displaystyle \sf{ = \frac{ (0 + \frac{ \sqrt{3} }{ 2} )( \frac{1}{2} + \frac{1}{ \sqrt{3}} + 1)}{( \frac{1}{ \sqrt{3} } + \frac{1}{ \sqrt{3} } )( 2 - \frac{2}{ \sqrt{3}} ) } }$

$\displaystyle \sf{ =\frac{ \frac{ \sqrt{3} }{ 2} ( \frac{3 \sqrt{3} + 2}{ 2\sqrt{3}} )}{\frac{2}{ \sqrt{3} }( \frac{2 \sqrt{3} - 2 }{ \sqrt{3}}) } }$

$\displaystyle \sf{ = \frac{3}{8} \times \frac{3 \sqrt{3} + 2}{2 \sqrt{3} - 2} }$

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