Question

Please prove it.......​

Answer 1

Answer:

For two real numbers x and y:

    (x - y)²  ≥  0

⇒  x² + y²  ≥  2xy

⇒  2(x² + y²)  ≥  (x + y)²

Putting x = sin θ and y = cos θ gives

    2(sin²θ + cos²θ)  ≥  (sin θ + cos θ)²

⇒  2  ≥  (sin θ + cos θ)²

⇒  √2  ≥  sin θ + cos θ.

So  sin θ + cos θ never takes a value larger than √2.

To see that this is actually the largest value, we must confirm that the value √2 actually occurs.

Putting  θ = π/4  gives  sin θ = cos θ = 1/√2,  and this gives

 sin θ + cos θ  =  2/√2 = √2.