Question

How to prove the value of cos a + sin a is greater than 1 ?

Answer 1

For `2kpi < a < pi/2+2kpi` for `k=0,1,2,cdots`

Explanation:

If `cos(a)+sin(a) > 1` then `(cos(a)+sin(a) )^2=1+2 cos(a)sin(a) > 1` or
`cos(a)sin(a) > 0`
Concluding it is sufficient `cos(a)sin(a) > 0` to verify
`cos(a)+sin(a) > 1`
or
`2kpi < a < pi/2+2kpi` for `k=0,1,2,cdots`

Attached a plot showing

`cos(a)+sin(a)` in blue
`max(cos(a)sin(a),0)` in red and
the constant `1` in green.

Answer 2

`sin a + cos a in [-sqrt 2, sqrt 2]`.

Explanation:

For `a in (-oo, oo)`,

`cos a + sin a=sqrt2 (1/sqrt2cos a+1/sqrt2 sin a)`

`=sqrt2(sin (pi/4)cos a+cos(pi/4)sin a`

`=sqrt2sin(a+pi/4) in [-sqrt2, sqrt2]`

As a is not stated to be in an interval, I can assign `a = -pi/4` to get

0 for sin a + cos a.