prove true or false: (1+cosA-sinA)^2 = 2(1+cosA).(1-cosA)

True.

(1 + cosA - sinA)² = 2(1 + cosA)(1 - cosA)

(1 + 2(cosA - sinA) + (cosA - sinA)² = 2(1 - cos²A)

(1 + 2(cosA - sinA) + (cos²A - 2sinAcosA + sin²A) = 2sin²A

1 + 2cosA - 2sinA + 1 - 2sinAcosA  = 2sin²A

2 + 2cosA - 2sinA - 2sinAcosA  = 2sin²A

1 + cosA - sinA - sinAcosA  = sin²A

Now if we put A = 45 we find LHS = RHS

Of course, we can put in the given expression directly any acute angle like 45, 60 or 30 and verify.

Regards,

Team,

TopperLearning.