# Find the exact value of tan 7π/ 8 by using a half-angle identity.

Tangent is a basic trigonometric function that can be defined as the ratio of opposite (perpendicular) side to the adjacent (base).

## Answer: The value of tan 7π/ 8 using half angle identity is equal to 1 - √2.

Let us find the value of tan 7π/ 8.

**Explanation**

Let, tan θ / 2 = tan 7π/ 8

⇒ Let, θ / 2 = 7π/ 8

Thus, θ = 7π/ 4

By using half angle identity of tangent function,

⇒ tan θ / 2 = sin θ / ( 1 + cos θ )

⇒ tan 7π/ 8 = (sin 7π/ 4) / (1 + cos 7π/ 4)

⇒ tan 7π/ 8 = sin (2π - π / 4) / [1 + cos (2π - π / 4)] --------------- (1)

By using the trigonometric formulae,

sin (A - B) = sin A cos B - cos A sin B

cos (A - B) = cos A cos B - sin A sin B

Using these in the RHS of equation (1) we get,

⇒ (sin 2π × cos π / 4 - cos 2π × sin π / 4) / [1 + (cos 2π × cos π / 4 + sin 2π × sin π / 4 )]

⇒ (0 × (1 / √2 ) - 1 × √2 ) / [1 + ( 1 × (1/ √2) + 0 × (1/ √2) )]

⇒ (- 1 / √2 ) / [1 + (1/ √2) + 0]

⇒ - 1 / (√2 + 1)

By rationalising the above expression we get,

⇒ - 1 / (√2 + 1) × (√2 - 1) / (√2 - 1)

⇒ - 1 (√2 - 1) / 2 -1

⇒ 1 - √2

### Thus, the value of tan 7π/ 8 is equal to 1 - √2.

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