त्रिकोणमितीय सारणी का प्रयोग किये बिना मान ज्ञात कीजिए :
`((tan 20^(@))/(cosec 70^(@)))^2 + ((cot 20^(@))/(sec 70^(@)))^2 + 2 tan 15^(@) tan37^(@)tan53^(@) tan 60^(@) tan75^(@)`

Explanation

`((tan 20^(@))/(cosec 70^(@))) + ((cot 20^(@))/(sec 70^(@))) + 2 tan 15^(@) tan37^(@)tan53^(@) tan 60^(@) tan75^(@)`
` = {(tan 20^(@)/("cosec " 90^(@) - 20^(@)))}^(2) + { ( cot 20^(@))/(sec(90^(@) - 20^(@)))}^(2)`
` + 2 tan 15^(@) tan 37^(@) tan (90^(@) - 37^(@))* (sqrt(3)) tan ( 90^(@) - 15^(@))`
`((tan 20^(@))/(sec 20^(@)))^(2)+((cot 20^(@))/(cosec 20^(@)))^(2) + 2 tan 15^(@) tan 37^(@) cot 37^(@) * (sqrt(3)) cot 15^(@)`
` = ((sin 20^(@)//cos 20^(@))/(1//cos 20^(@)))^(2) + ((cos 20^(@)//sin 20^(@))/(1//sin 20^(@)))^(2) + 2sqrt(3) tan 15^(@) tan 37^(@)* (1)/( tan 37^(@)) * (1)/(tan 15^(@))`
` = sin^(2) 20 ^(@) + cos^(2) 20^(@) + 2sqrt(3) = 1 + 2sqrt(3)`