The value of the expression 1 + c o s e c π 4 + c o s e c π 8 + c o s e c π 16 is equal to

Step-by-step solution

We know that c o s e c θ + cot ⁡ θ = 1 + cos ⁡ θ sin ⁡ θ = 2 c o s 2 θ 2 2 s i n θ 2 c o s θ 2 = c o t θ 2 ⇒ c o s e c θ = c o t θ 2 - cot ⁡ θ ⇒ c o s e c π 4 = c o t π 8 - c o t π 4 c o s e c π 8 = c o t π 16 - c o t π 8 c o s e c π 16 = c o t π 32 - c o t π 16 Adding them, we get, c o s e c π 4 + c o s e c π 8 + c o s e c π 16 = c o t π 32 - c o t π 4 ⇒ 1 + c o s e c π 4 + c o s e c π 8 + c o s e c π 16 = c o t π 32