Question

if cos theta + sin theta = 1, then prove that cos theta - sin theta = +- 1

Answer 1

cos theta + sin theta = 1 --------(1)

(a + b)^2 + (a - b)^2 = a^2 + b^2 + 2ab + a^2 + b^2 - 2ab

(a+b)^2 + (a-b)^2 = 2a^2 + 2b^2

=> (a+b)^2 + (a-b)^2 = 2 ( a^2 + b^2)

Similarly,

( sin theta + cos theta)^2 + (cos theta - sin theta)^2 = 2 ( sin^2 theta + cos^2 theta)

=> ( 1)^2 + (cos theta - sin theta)^2 = 2 ( 1) [ using equation 1]

=> 1 + ( cos theta - sin theta)^2 = 2

=> ( cos theta - sin theta)^2 = 1

=> cos theta - sin theta = + - 1

Hence Proved...