If sin ⁡ A and cos ⁡ A are the roots of the equation 4 x 2 - 3 x + a = 0 , sin ⁡ A + cos ⁡ A + tan ⁡ A + cot ⁡ A + sec ⁡ A + cosec A = 7 and 0 < A < π 2 , then the value of a must be

Step-by-step solution

sin ⁡ A and cos ⁡ A are the roots of the equation 4 x 2 - 3 x + a = 0 ⇒ sin ⁡ A + cos ⁡ A = 3 4 and sin ⁡ A cos ⁡ A = a 4 Also, sin ⁡ A + cos ⁡ A + tan ⁡ A + cot ⁡ A + sec ⁡ A + c o s e c A = 7 ⇒ sin ⁡ A + cos ⁡ A + 1 sin ⁡ A cos ⁡ A + sin ⁡ A + cos ⁡ A sin ⁡ A cos ⁡ A = 7 ⇒ 3 4 + 1 a 4 + 3 4 a 4 = 7 ⇒ 3 4 + 4 a + 3 a = 7 ⇒ 7 a = 7 - 3 4 = 25 4 ⇒ a = 28 25