if A+B=45 degree show that (cos A-1)(cotB-1)=2.

To prove that (cos A - 1)(cot B - 1) = 2 given A + B = 45 degrees, we will use trigonometric identities and properties. Here's the step-by-step explanation:

Step 1: Express cot B in terms of sin and cos:
cot B = cos B / sin B

Step 2: Rewrite the given expression using the identity for cos A - 1:
(cos A - 1)(cot B - 1)
= (cos A - 1)(cos B / sin B - 1)

Step 3: Change the denominator of the second term to have a common denominator of sin B:
= (cos A - 1)((cos B - sin B) / sin B)

Step 4: Rewrite (cos B - sin B) as √2 sin (B - 45):
= (cos A - 1)(√2 sin (B - 45) / sin B)

Step 5: Simplify the expression by dividing sin (B - 45) by sin B:
= (cos A - 1)(√2 sin B cos 45 - cos B sin 45) / sin^2 B

Step 6: Simplify sin 45 and cos 45 (√2 / 2):
= (cos A - 1)(√2 sin B (√2 / 2) - cos B (√2 / 2)) / sin^2 B

Step 7: Further simplify by canceling out (√2 / 2):
= (cos A - 1)(sin B - cos B√2) / sin^2 B

Step 8: Use the trigonometric identity sin^2 A + cos^2 A = 1:
= (cos A - 1)((sin B - cos B√2) / (1 - cos^2 B))

Step 9: Substitute sin^2 B = 1 - cos^2 B using the Pythagorean identity:
= (cos A - 1)((sin B - cos B√2) / (1 - (1 - sin^2 B)))

Step 10: Simplify the denominator:
= (cos A - 1)((sin B - cos B√2) / sin^2 B)

Step 11: Substitute A + B = 45 degrees (given):
= (cos (45 - B) - 1)((sin B - cos B√2) / sin^2 B)

Step 12: Simplify cos (45 - B) using the identity cos (90 - θ) = sin θ:
= (sin B - 1)((sin B - cos B√2) / sin^2 B)

Step 13: Simplify the expression further:
= sin^2 B - sin B + cos B√2 - √2 sin B + √2 cos B - √2

Step 14: Apply the Pythagorean identity sin^2 B + cos^2 B = 1:
= 1 - sin B + cos B√2 - √2 sin B + √2 cos B - √2

Step 15: Combine like terms:
= -sin B - √2 sin B + √2 cos B + cos B - √2 - √2

Step 16: Rearrange the terms:
= √2 cos B + cos B - sin B - √2 sin B - √2 - √2

Step 17: Factor out √2:
= (√2 + 1)(cos B - sin B) - 2(√2 + 1)

Step 18: Simplify (√2 + 1) - 2(√2 + 1):
= (√2 + 1)(cos B - sin B - 2)

Step 19: Notice that A + B = 45 degrees implies (90 - B) + B = 45 degrees, which simplifies to 90 degrees = 45 degrees, which is not possible. Therefore, there must be an error or inconsistency in the given information or expression.

In conclusion, the expression (cos A - 1)(cot B - 1) does not equal 2 when A + B = 45 degrees.