Answer:
To verify the given identity:
(1 - cos a) (1 + cot a)
= (1 - cos a) (1 + cos a / sin a) [since cot a = cos a / sin a]
= 1 - cos^2 a / sin a + cos a - cos^2 a / sin a
= 1 - (cos^2 a + cos^2 a) / sin a + cos a
= 1 - 2 cos^2 a / sin a + cos a
= 1 - 2 (1 - sin^2 a) / sin a + cos a [since cos^2 a = 1 - sin^2 a]
= 1 - 2 / sin a + 2 sin a / sin a + cos a
= 1 - 2 / sin a + 2 + cos a
= 1 + 2 (1 - sin a) / sin a
= 1 + 2 cos^2 a / sin a
= 1 + 2 cot^2 a
= (1 + cot^2 a) + 2 cot^2 a
= cosec^2 a + 2 cot^2 a
= 1 + cot^2 a [since cosec^2 a = 1 + cot^2 a]
Therefore, (1 - cos a) (1 + cot a) = 1 is true.