tan (45° + θ) – cot (45° – θ) is equal to –2cos θ02sin θ1

The given expression is tan (45° + θ) – cot (45° – θ).

Let's simplify this step by step:

Step 1: Convert the given expression into sine and cosine terms.

tan (45° + θ) = sin (45° + θ) / cos (45° + θ)

cot (45° – θ) = cos (45° – θ) / sin (45° – θ)

Step 2: Use the formula for sin (A + B) and cos (A + B) to expand the terms.

sin (45° + θ) = sin 45° cos θ + cos 45° sin θ = (1/√2)(cos θ + sin θ)

cos (45° + θ) = cos 45° cos θ - sin 45° sin θ = (1/√2)(cos θ - sin θ)

cos (45° – θ) = cos 45° cos θ + sin 45° sin θ = (1/√2)(cos θ + sin θ)

sin (45° – θ) = sin 45° cos θ - cos 45° sin θ = (1/√2)(cos θ - sin θ)

Step 3: Substitute these values back into the original expression.

tan (45° + θ) – cot (45° – θ) = [(1/√2)(cos θ + sin θ) / (1/√2)(cos θ - sin θ)] - [(1/√2)(cos θ + sin θ) / (1/√2)(cos θ - sin θ)]

Step 4: Simplify the expression.

This simplifies to 0, which is not equal to -2cos θ02sin θ1.

Therefore, the given expression tan (45° + θ) – cot (45° – θ) is not equal to -2cos θ02sin θ1.