सिद्ध कीजिए कि -
(i) `sin 40^(@) cos 50^(@) + cos 40^(@) sin 50^(@) = 1 `
(ii) ` cot ( 270^(@) - theta ) cot ( 270^(@) + theta ) cot ( 540^(@) - theta ) cot ( 540^(@) + theta ) = 1 ` .
(iii) ` cos. (pi)/8 + cos. (3pi)/8 + cos . (5 pi)/8 + cos . (7 pi)/8 = 0 .`

Explanation

(i) ` L.H.S = sin 40^(@) cos 50^(@) + cos 40^(@) sin 50^(@)`
` = sin 40 ^(@) cos ( 90^(@) - 40^(@))`
` + cos 40^(@) sin (90^(@)-40^(@))`
` = sin^(2) 40^(@) + cos^(2) 40^(@)`
` = 1 = R.H.S`
(ii) ` L.H.S = cot ( 270^(@) - theta) cot (270^(@) + theta ) cot (540^(@) - theta ) `
` cot ( 540^(@) + theta) `
` = tan theta ( - tan theta) * cot { 36-0^(@) + (180^(@)-theta)}`
` * cot { 360^(@) + ( 180^(@) + theta)}`
` = - tan^(2) theta * ( -cot theta ) ( cot theta) `
` = -1/(cot^(2) theta ) ( - cot^(2) theta)`
` = 1 = R.H.S`
(iii) ` L.H.S = cos. pi/8 + cos. (3pi)/8 + cos . (5pi)/8 + cos. (7pi)/8 `
` = cos. pi/8 + cos. (3pi)/8 `
` + cos. (pi - (3pi)/8) + cos(pi-pi/8)`
` = cos. pi/8 + cos. (3pi)/8 - cos. (3pi)/8 - cos. pi/8`
` = 0 = R.H.S`