Question

If tan  π/9, x, tan 15π/18 are in A.P and tan π/9, tan 7π/18 are in A.P, then 

(a)  2x = y

(b)  x = y

(c)  x = 2y

(d)  None of these 

Answer 1

Correct option  (a) 2x = y

Explanation:

2x = tan20° +  tan50° 

2y = tan20° + tan70°

2x = sin 20° .cos50° +  cos 20° .sin 50°/cos 20° .cos 70°

2y = sin 20° .cos 70° cos 20° .sin 70°/cos 20° .cos 70°

2x = sin 70°/cos50° .cos 20°

2y = sin 90°/sin 20° .cos 20° 

2x = cos 20°/cos50° .cos 20°

2y = sin 90°/sin20° .cos 20°

 2x = cos 20°/cos50° .cos 20°

⇒ 2x = 1/cos50° 1/ sin 40° ...(1)

2y =2/ sin 40° .....................(2)

From (1) and (2)

2x = y