If sin ⁡ x + cos ⁡ x = 7 2 , where x ∈ 0 , π 4 , then the value of t a n x 2 is equal to

Step-by-step solution

sin ⁡ x + cos ⁡ x = 7 2 ⇒ 2 t a n x 2 1 + t a n 2 x 2 + 1 - t a n 2 x 2 1 + t a n 2 x 2 = 7 2 ⇒ 4 t a n x 2 + 2 - 2 t a n 2 x 2 = 7 + 7 t a n 2 x 2 ⇒ 2 + 7 t a n 2 x 2 - 4 t a n x 2 + 7 - 2 = 0 ⇒ t a n x 2 = 4 ± 16 - 4 2 + 7 7 - 2 2 2 + 7 = 4 ± 16 - 4 × 3 2 2 + 7 tan x 2 = 2 ± 1 2 + 7 = 1 2 + 7   or   3 2 + 7 t a n x 2 = 7 - 2 3 or 7 - 2 But, as x 2 ∈ 0 , π 8 ⇒ t a n x 2 = 7 - 2 3