Calculate the value tan 9° - tan 27°- tan 63° + tan 81°

The trigonometric functions are defined as the functions of an angle of a triangle which is also known as circular functions. The trigonometric functions give the relationship between the angles and sides of a triangle. The basic trigonometric functions are sine, cosine, tangent, cotangent, secant and cosecant.

Answer: The value tan 9° - tan 27°- tan 63° + tan 81° is 4.

We will use the trigonometric ratio of tan(90° - θ) with few steps to calculate the value tan 9° - tan 27°- tan 63° + tan 81°.

Explanation:

tan 9° – tan 27°- tan 63° + tan 81° = tan 9° + tan 81° - tan 27° - tan 63°

Apply the trigonometric formula: tan (90° - θ) = cot θ

tan 9° + tan 81° - tan 27° - tan 63° = tan 9° + tan (90°- 9°) – tan 27° – tan (90°- 27°)

= tan 9° + cot 9° – tan 27° – cot 27°

= tan 9° + cot 9° - (tan 27° + cot 27°) --- (1)

Using identities, 2 cosA sinA = sin 2A, tan A = sin A/cos A, cot A = cos A/sin A, sin² A + cos² A = 1

We can express tan 9° + cot 9° as tan 9°+ cot 9° = (sin² 9° + cos² 9°)/ (sin 9°cos 9°) = 2 / sin 18° --- (2)

tan 27° + cot 27° = (sin² 27° + cos² 27°) / (sin 27° cos 27°) = 2 / sin 54° = 2 / cos 36° --- (3)

Substitute the values of (2) and (3) in (1), we get

tan 9° + cot 9° - (tan 27° + cot 27°) = (2 / sin 18° ) - (2 / cos 36°) = [(2×4)/(√5-1)] - [(2×4)/(√5+1)] = 4

Hence, the value tan 9° - tan 27°- tan 63° + tan 81° is 4.