Given:
cot 333 - cos 567/tan 297 * sin 477

To prove:
cot 333 - cos 567/tan 297 * sin 477 = 1

Proof:

To simplify the given expression, we will break it down into smaller parts and simplify each part step by step.

Step 1: Simplify cot 333
The cotangent function is the reciprocal of the tangent function.
cot 333 = 1 / tan 333

Step 2: Simplify cos 567
The cosine function is periodic with a period of 360 degrees (or 2π radians). Therefore, cos 567 is equivalent to cos (567 - 360) = cos 207.

Step 3: Simplify tan 297
The tangent function is periodic with a period of 180 degrees (or π radians). Therefore, tan 297 is equivalent to tan (297 - 180) = tan 117.

Step 4: Simplify the expression further
Now, we can substitute the simplified values into the expression:
cot 333 - cos 567/tan 297 * sin 477
= 1/tan 333 - cos 207/tan 117 * sin 477

Step 5: Simplify the remaining trigonometric functions
We can use the trigonometric identities to simplify the expression further:
1/tan 333 = cot 333 (since cot 333 = 1/tan 333)
cos 207/tan 117 = cot 117 * cos 207 (since cot θ = 1/tan θ)
cot 117 * cos 207 * sin 477 = cot 117 * sin 477 * cos 207 (since sin θ * cos φ = sin φ * cos θ)

Step 6: Simplify cot 117 * sin 477 * cos 207
cot 117 is the reciprocal of tan 117, so cot 117 = 1/tan 117
cot 117 * sin 477 * cos 207 = 1/tan 117 * sin 477 * cos 207

Step 7: Use trigonometric identities to simplify further
We can use the identity sin 2θ = 2sin θ * cos θ to simplify the expression:
1/tan 117 * sin 477 * cos 207 = 1/tan 117 * sin (2 * 477) * cos 207
= 1/tan 117 * 2sin 477 * cos 477 * cos 207

Step 8: Simplify the expression once again
Using the identity cos (θ + φ) = cos θ * cos φ - sin θ * sin φ, we can simplify further:
1/tan 117 * 2sin 477 * cos 477 * cos 207 = 1/tan 117 * 2sin 477 * (cos (477 + 207))
= 1/tan 117 * 2sin 477 * (cos 477 * cos 207 - sin 477 * sin 207)

Step 9: Simplify the expression to obtain the desired result