Find tan 22.5 degree using the Half-angle Formula?

In a right-angled triangle, the tangent of an angle is defined as the ratio of length of the side opposite to the angle to the length of the adjacent side.

Answer: The value of tan 22.5 degree is √2 – 1

Let us proceed step by step

Explanation:

From trigonometric formulae,

tan 2x = 2 tanx / ( 1 – tan²x) ---------------- (1)

Here, x = 22.5 °

Therefore, 2x = 2 × 22.5° = 45°

Also, we know that tan 45° = 1 (from trigonometric table values)

Let us consider tan 22.5° = y

Substituting x = 22.5 ° and tan 22.5° = y in (1) we get,

⇒1 = 2y / (1 – y²)

⇒1 – y² = 2y

⇒ y² + 2y – 1 = 0

⇒ y = [- 2 ± √(2² – 4 (1) (-1) ] / 2 [Using quadratic formula]

⇒ y = ( – 2 ± √8 ) / 2

⇒ y = ( – 2 ± 2√2 ) / 2 

⇒ y = – 1 ± √2    [dividing numerator and denominator by 2]

⇒ y = √2 – 1 or

⇒ y = -1 – √2

Since the value of tan 22.5 degrees lies in the 1st quadrant, therefore, the required value should be positive.

Therefore, the value of tan 22.5 degree is √2 – 1.