Question 1173666
<pre>

I realized there was a simpler solution without extraneous solutions:

{{{2}}}{{{""=""}}}{{{sqrt(2)cos^""(expr(1/2)(x-30^o))+1}}} 

Subtract 1 from both sides:

{{{1}}}{{{""=""}}}{{{sqrt(2)cos^""(expr(1/2)(x-30^o))}}}

Divide both sides by √2

{{{1/sqrt(2)}}}{{{""=""}}}{{{cos^""(expr(1/2)(x-30^o))}}}

{{{sqrt(2)/2}}}{{{""=""}}}{{{cos^""(expr(1/2)(x-30^o))}}}

{{{expr(1/2)(x-30^o)}}}{{{""=""}}}{{{"" +- 45^o +n*360^o}}}

Multiply both sides by 2

{{{x-30^o}}}{{{""=""}}}{{{"" +- 90^o +n*720^o}}}

Add 30° to both sides

{{{x}}}{{{""=""}}}{{{30^o +- 90^o +n*720^o}}}

Since we are given:

{{{-180^o<= x<= 360^o}}}

{{{-180^o<=30^o +- 90^o +n*720^o <=360^o}}}

We subtract 30° from all three sides

{{{-210^o<="" +- 90^o +n*720^o <=330^o}}}

We subtract or add 90° from/to all three sides:

{{{-210^o +- 90^o<=n*720^o <=330^o +- 90^o}}}

Divide all three sides by 720°

{{{-210^o/720^o +- 90^o/720^o<=n <=330^o/720^o +- 90^o/720^o}}}

{{{-7/24 +- 1/8<=n<=11/24 +- 1/3}}}

Using the +

{{{-7/24 + 1/8<=n<=11/24 + 1/3}}}

{{{-1/6 <= n <= 19/24}}}

Since n is an integer, n=0

{{{x}}}{{{""=""}}}{{{30^o + 90^o +0*720^o}}}

{{{x}}}{{{""=""}}}{{{120^o}}}

Using the -

{{{-7/24 - 1/8<=n<=11/24 - 1/3}}}

{{{-5/12 <= n <= 1/8}}}

Since n is an integer, n=0

{{{x}}}{{{""=""}}}{{{30^o - 90^o +0*720^o}}}

{{{x}}}{{{""=""}}}{{{-60^o}}}

There are two solutions:

-60° and 120° 

Edwin</pre>