If s i n 23 π 24 = 2 p - q - 1 4 r , then the value of p 2 + q 2 - r 2 is equal to

Step-by-step solution

s i n 23 π 4 = s i n π - π 24 = s i n π 24 Now, c o s π 12 = cos ⁡ 15 o = 3 + 1 2 2 Using 1 - c o s  θ = 2 s i n 2 θ 2 , we get 2 s i n 2 π 24 = 1 - c o s π 12 = 1 - 3 + 1 2 2 ⇒ s i n 2 π 24 = 2 2 - 3 - 1 4 2 ⇒ s i n π 24 = s i n 23 π 24 = 2 2 - 3 - 1 4 2 ⇒ p = 2 , q = 3 , r = 2 ⇒ p 2 + q 2 - r 2 = 9