To solve the given equation, let's simplify it step by step:

Step 1: Simplify the expression inside the first inverse tangent (tan^-1).

tan^-1 (sin^2θ + 2sinθ + 2)

Step 2: Use the trigonometric identity sin^2θ + cos^2θ = 1.

sin^2θ + 2sinθ + 2 = 1 - cos^2θ + 2sinθ + 2

Step 3: Rearrange the terms.

cos^2θ - 2sinθ + 1 = 0

Step 4: Factorize the quadratic equation.

(cosθ - 1)(cosθ + 1) - 2sinθ = 0

Step 5: Simplify.

(cosθ - 1)(cosθ + 1) = 2sinθ

Step 6: Divide both sides by cosθ.

cosθ + 1 = 2sinθ / cosθ

Step 7: Use the trigonometric identity tanθ = sinθ / cosθ.

cosθ + 1 = 2tanθ

Step 8: Simplify.

2tanθ - cosθ - 1 = 0

Now, let's simplify the expression inside the second inverse cotangent (cot^-1).

cot^-1 (4sec^2φ + 1)

Step 1: Use the trigonometric identity sec^2φ = 1 + tan^2φ.

4(1 + tan^2φ) + 1

Step 2: Simplify.

4 + 4tan^2φ + 1

Step 3: Combine like terms.

5 + 4tan^2φ

Step 4: Use the trigonometric identity tan^2φ = sec^2φ - 1.

5 + 4(sec^2φ - 1)

Step 5: Simplify.

4sec^2φ + 1

Now, let's substitute the simplified expressions back into the original equation:

2tanθ - cosθ - 1 = cot^-1 (4sec^2φ + 1)

Since the equation is equal to π/2, we can rewrite it as:

2tanθ - cosθ - 1 = π/2

Now, let's analyze the options and determine which one(s) satisfy the equation:

A) sinθ = -1
If sinθ = -1, then tanθ = sinθ / cosθ = -1 / cosθ.
Substituting into the equation:
2(-1 / cosθ) - cosθ - 1 = π/2
This equation does not hold true for all values of cosθ, so option A is not a valid solution.

B) sinφ = 1
If sinφ = 1, then cosφ = √(1 - sin^2φ) = √(1 - 1) = 0.
Substituting into the equation:
2tanθ - cosθ - 1 = π/2
This equation does not hold true for all values of tanθ, so option B is not a valid solution.

C) cosθ = 1
If cosθ = 1, then tanθ = sinθ / cosθ = sinθ.
Substituting into the equation:
2sinθ - 1 -