Tan^4 pi/8 - 4tan^2 pi/8 +4tan pi/8 + 1pls fast ...email on ssoni0097gmail.com

Let's solve the given expression step by step:

Expression:
tan⁴(π/8) - 4tan²(π/8) + 4tan(π/8) + 1

Step 1: Let tan(π/8) = x
Substituting x in the given expression:

x⁴ - 4x² + 4x + 1

Step 2: Using the identity for tan(π/8)
We use the identity:

tan(π/8) = √(1 - cos(π/4)) / √(1 + cos(π/4))

Since cos(π/4) = 1/√2, we get:

tan(π/8) = √(1 - 1/√2) / √(1 + 1/√2)

Simplifying:

tan(π/8) = √((√2 - 1)/√2) / √((√2 + 1)/√2)

Using further simplifications, we get:

tan(π/8) = (√2 - 1)

So, let x = (√2 - 1)

Step 3: Compute x²
x² = (√2 - 1)²
= (2 - 2√2 + 1)
= 3 - 2√2

Step 4: Compute x⁴
x⁴ = (3 - 2√2)²
= 9 - 12√2 + 8
= 17 - 12√2

Step 5: Compute the expression
Substituting x⁴ and x² in the original equation:

(17 - 12√2) - 4(3 - 2√2) + 4(√2 - 1) + 1

Expanding:

= 17 - 12√2 - 12 + 8√2 + 4√2 - 4 + 1
= (17 - 12 - 4 + 1) + (-12√2 + 8√2 + 4√2)
= 2 + 0
= 2

Final Answer:
The value of the given expression is 2.