Prove that

Options

A. `2 " sin " 22 .(1^(@))/(2) cos 22 .(1^(@))/(2)=(1)/(sqrt(2))`
B. `2 cos^(2) 15^(@) -1 =(sqrt(3))/(2)`
C. `8 cos^(3) 20^(@) -6 cos 20^(@) =1`
D. `3 sin 40^(@) - 4 sin^(3) 40^(@) = (sqrt(3))/(2)`

Explanation

(i) LHS = `sin (2 xx 22 .(1^(@))/(2))= sin 45^(@) =(1)/(sqrt(2))`
(ii) LHS= `=(2 cos^(2) theta -1) = cos 2theta " where " theta = 15^(@)`
= cos `30^(@) = (sqrt(3))/(2)`
(iii) LHS = `2(4 cos^(2) theta -1) = " cos " 2 theta " where " theta =20^(@) `
`=2 xx cos 3theta = 2 xx cos 60^(@) = (2xx .(1)/(2))=1`
(iv) `LHS = (3 sin theta -4 sin^(3) theta) " where " theta =40^(@)`
`=sin 3theta =sin 120^(@) = sin 60^(@) = (sqrt(3))/(2)`