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What is  the value of \(\frac{{tan9^\circ tan23^\circ tan60^\circ tan67^\circ tan81^\circ }}{{cosec{^2}72^\circ + {{\cos }^2}15^\circ - {{\tan }^2}18^\circ + {{\cos }^2}75^\circ }} \)

What is  the value of \(\frac{{tan9^\circ tan23^\circ tan60^\circ tan67^\circ tan81^\circ }}{{cosec{^2}72^\circ + {{\cos }^2}15^\circ - {{\tan }^2}18^\circ + {{\cos }^2}75^\circ }} \) Correct Answer <span class="math-tex">\(\frac{\sqrt 3}{2}\)</span>

Formulae used :

[ alt="Trigo - Formula" src="//storage.googleapis.com/tb-img/production/21/02/Trigo%20-%20Formula.PNG" style="width: 450px; height: 202px;">

tanθ = 1/cotθ, Cosec2θ – cot2θ = 1  and cos2θ  + sin2θ = 1

[ alt="Trigo - Angles and Radian" src="//storage.googleapis.com/tb-img/production/21/02/Trigo%20-%20Angles%20and%20Radian.PNG" style="width: 392px; height: 218px;">

Calculations :

tan 9° tan 23° tan 60° tan 67° tan 81°/cosec272° + cos215° - tan218° + cos275° 

= cot81° cot67° tan60° tan67° tan81°/cosec272 - cot272° + cos215° + sin215° 

= tan60°/(1 + 1) 

= √3/2 

∴ The required value will be √3/2 

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