निम्नलिखित के मान ज्ञात करें ।
(i) ` sin 48^(@) * sec 42^(@) + cos 48^(@) * cosec 42^(@)`
(ii) `tan 35^(@) * tan 40^(@) * tan 45^(@) * tan 50^(@) * tan 55^(@)`
(iii) ` (cos 75^(@))/(sin 15^(@)) + (sin 12^(@))/(cos 78^(@)) - (cos 18^(@))/(sin 72^(@)) `
(iv) `((sin 47^(@))/(cos 43^(@)))^(2) + ((cos 43^(@))/(sin 47^(@)))^(2) - 2 cos^(2) 45^(@)`

Explanation

(i) ` sin 48^(@) * sec 42^(@) + cos 48^(@) * cosec 42^(@)`
` = sin 48^(@) * sec (90^(@) - 48^(@)) + cos 48^(@) cosec (90^(@) - 48^(@))`
` = sin 48^(@) * cosec 48^(@) + cos 48^(@) * sec 48^(@)`
` = 1 + 1 = 2" "[:' sin theta xx cosec theta = 1 , cos theta . sec theta = 1]`
(ii) ` tan 35^(@) * tan 40^(@) * tan 45^(@) * tan 50^(@) * tan 55^(@)`
` = tan 35^(@) * tan 40^(@) * 1 . tan (90^(@) - 40^(@)). tan (90^(@) - 35^(@))`
` = tan 35^(@) * tan 40^(@) * cot 40^(@) * cot 35^(@)`
` = tan 35^(@) * cot 35^(@) * tan 40^(@) * cot 40^(@) =1" "[:' tan theta xx cot theta = 1]`
(iii)` (cos 75^(@))/(sin 15^(@)) + (sin 12^(@))/(cos 78^(@)) - (cos 18^(@))/(sin 72^(@))`
` = (cos 75^(@))/(sin (90^(@) - 75^(@)))+ (sin 12^(@))/(cos (90^(@) - 12^(@))) - (cos 18^(@))/(sin (90^(@) - 18^(@)))`
` = (cos 75^(@))/(cos 75^(@)) + (sin 12^(@))/(sin 12^(@)) - (cos 18^(@))/(cos 18^(@)) = 1 + 1 - 1 = 1`
(iv) ` ((sin 47^(@))/(cos 43^(@)))^(2) + ((cos 43^(@))/(sin 47^(@)))^(2) - 2 cos^(2) 45^(@)`
` = [(sin 47^(@))/(cos (90^(@) - 47^(@)))]^(2) + [(cos 43^(@))/(sin (90^(@)-43^(@)))]^(2) - 2 cos^(2) 45^(@)`
` = ((sin 47^(@))/(sin 47^(@)))^(2) + ((cos 43^(@))/(cos 43^(@)))^(2) - 2 xx (1/sqrt2)^(2) = 1 + 1 - 2 xx 1/2 = 1`