sin (x + p/4) + sin (x - p/4) = -1  becomes

sin x cos p/4 + sin p/4 cos x + sin x cos p/4 - cos x sin p/4 = -1

Putting in values for cos p/4 and sin p/4,

(sin x) (\/2  /2) + ((\/2  /2)(cos x) + sin x (\/2  /2) - (cos x) (\/2  /2) = -1

Combining like terms,

2(sin x) (\/2  /2) = -1

(sin x) (\/2  /2) = -1/2

sin x = -1/\/2   and

x = -p/4 + pn