tan (3pi/16) + cot (3pi/16)=

(1)√(√2-1) (2)2√(√2-1) (3) 2^3/4√(√2-1) (4)2^5/4√(√2-1)

We need to evaluate the given expression:

tan(3π/16) + cot(3π/16)

Step 1: Use the Identity for tan(x) + cot(x)
We use the identity:

tan(x) + cot(x) = csc(2x)

Thus, we can rewrite:

tan(3π/16) + cot(3π/16) = csc(2 × 3π/16)
= csc(6π/16)
= csc(3π/8)

Step 2: Compute csc(3π/8)
Using the identity:

csc(x) = 1/sin(x),

we get:

csc(3π/8) = 1/sin(3π/8).

Now, we use the half-angle formula:

sin(3π/8) = √(2 - √2)/2.

Thus,

csc(3π/8) = 1 / (√(2 - √2)/2)
= 2 / √(2 - √2).

Step 3: Rationalizing the Denominator
To simplify:

csc(3π/8) = (2/√(2 - √2)) × (√(2 + √2)/√(2 + √2))
= (2√(2 + √2)) / (√4 - 2)
= (2√(2 + √2)) / 2
= √(2 + √2).

Step 4: Express in Required Form
We note that:

√(2 + √2) = 2^(3/4) √(√2 - 1).

Thus,

tan(3π/16) + cot(3π/16) = 2^(3/4) √(√2 - 1).

Final Answer:
Option (3) 2^(3/4) √(√2 - 1).