Let α be the angle between the lines whose direction cosines satisfy the equations l + m - n = 0 and l 2 + m 2 - n 2 = 0 . Then the value of sin 4 α + cos 4 α is :

Step-by-step solution

n = ℓ + m Now, ℓ 2 + m 2 = n 2 = ℓ + m 2 ⇒   2 ℓ m = 0 l 2 + m 2 + n 2 = 1 If ℓ = 0 ⇒   2 n 2 = 1 ⇒ n = ± 1 2 m = n = ± 1 2 And, If m = 0 ⇒ n = ℓ = ± 1 2 ∴   l 2 + m 2 = 1 2 &   l + m = 1 2 ⇒ 1 2 + 2   lm = 1 2 ∴   l = 0 ,   m = 1 2 or l = 1 2 ,   m = 0 So, direction cosines of two lines are 0 , 1 2 , 1 2 and 1 2 , 0 , 1 2 Thus, ∴   cos   α = 0 + 0 + 1 2 = 1 2