If the value of the expression tan ⁡ 1 2 cos - 1 ⁡ 2 5 is in the form of a + b where a , b ∈ Z , then the value of a + b b is K, Find 10K.

Step-by-step solution

We know that, tan ⁡ x 2 = 1 - cos ⁡ x sin ⁡ x ∴ tan ⁡ 1 2 cos - 1 ⁡ 2 5 = 1 - cos ⁡ cos - 1 ⁡ 2 5 sin ⁡ cos - 1 ⁡ 2 5 = 1 - 2 / 5 sin ⁡ sin - 1 ⁡ 1 / 5 = 1 - 2 / 5 1 / 5 = 5 - 2 Now, a + b = − 2 + 5 ⇒ a = − 2 and b = 5 Hence, a + b b = 3 5 = 0.6