please help me solve these 2 problems. I have no idea how to tackle them. Thank you.

Question

Using the facts that sine (-beta)= sine beta and cosine(-beta)=cosines beta to prove that sine alpha cosine alpha is less than 1/2

 

prove that sine alpha+ cosine alpha <= radical 2

Answer 1

sin (-β) = -sin β.

 

(sin α)(cos α) ≤ 1/2

2 sin α cos α ≤ 2

sin 2α ≤ 2

True conclusion

 

sin α + cos α ≤ √2

sin² α + 2 sin α cos α + cos² α ≤ 2

1 + 2 sin α cos α ≤ 2

2 sin α cos α ≤ 1

sin 2α ≤ 1

True conclusion