\(\displaystyle sin^6(x)+cos^6(x) \ = \ (7/12)sin^2(2x)\)
\(\displaystyle sin^6(x)+[cos^2(x)]^3 \ = \ (7/12)[2sin(x)cos(x)]^2\)
\(\displaystyle sin^6(x)+[1-sin^2(x)]^3 \ = \ (7/12)[4sin^2(x)cos^2(x)]\)
\(\displaystyle sin^6(x)+1-3sin^2(x)+3sin^4(x)-sin^6(x) \ = \ (7/3)[sin^2(x)cos^2(x)]\)
\(\displaystyle 1-3sin^2(x)+3sin^4(x) \ = \ (7/3)[sin^2(x)(1-sin^2(x)]\)
\(\displaystyle 1-3sin^2(x)+3sin^4(x) \ = \ (7/3)[sin^2(x)-sin^4(x)]\)
\(\displaystyle 1-3sin^2(x)+3sin^4(x) \ = \ (7/3)sin^2(x)-(7/3)sin^4(x)\)
\(\displaystyle 3sin^4(x)+(7/3)sin^4(x)-3sin^2(x)-(7/3)sin^2(x)+1 \ = \ 0\)
\(\displaystyle (16/3)sin^4(x)-(16/3)sin^2(x)+1 \ = \ 0\)
\(\displaystyle sin^4(x)-sin^2(x)+(3/16) \ = \ 0\)
\(\displaystyle Quadratic: \ sin^2(x) \ = \ \frac{1\pm.5}{2} \ = \ \frac{3}{4},\frac{1}{4}\)
\(\displaystyle Hence, \ sin(x) \ = \ \pm\frac{\sqrt3}{2},\pm\frac{1}{2}\)
\(\displaystyle Ergo, \ x \ = \ arcsin\bigg(\pm\frac{\sqrt3}{2}\bigg), \ x \ = \ arcsin\bigg(\pm\frac{1}{2}\bigg)\)
\(\displaystyle Thus, \ x \ = \ \pm\pi/3 \ or \ x \ = \ \pm\pi/6\)
\(\displaystyle Now, \ sin(x) \ = \ 1/2, \ then;\)
\(\displaystyle x \ = \ \pi/6+2k\pi \ or \ 5\pi/6+2k\pi \ or \ 7\pi/6+(2k-1)\pi \ or \ 11\pi/6+(2k-1)\pi, \ k \ an \ integer.\)
\(\displaystyle sin(x) \ = \ -1/2, \ then;\)
\(\displaystyle x \ = \ \pi/6+(2k-1)\pi \ or \ 5\pi/6+(2k-1)\pi \ or \ 7\pi/6+2k\pi \ or \ 11\pi/6+2k\pi.\)
\(\displaystyle sin(x) \ = \ \sqrt3/2, \ then;\)
\(\displaystyle x \ = \ \pi/3+2k\pi \ or \ 2\pi/3+2k\pi \ or \ 4\pi/3+(2k-1)\pi \ or \ 5\pi/3+(2k-1)\pi.\)
\(\displaystyle Anf \ finally \ if \ sin(x) \ = \ -\sqrt3/2, \ then;\)
\(\displaystyle x \ = \ \pi/3+(2k-1)\pi \ or \ 2\pi/3+(2k-1)\pi \ or \ 4\pi/3+2k\pi \ or \ 5\pi/3+2k\pi.\)
\(\displaystyle All \ in \ all, \ for \ (x), \ there \ are \ infinite \ possible \ solutions \ in \ real \ number \ land.\)
\(\displaystyle However, \ if \ we \ restrict \ x \ to \ 0 \ \le \ x \ \le \ 2\pi, \ then \ we \ have:\)
\(\displaystyle x \ = \ \pi/6, \ 5\pi/6, \ 7\pi/6, \ 11\pi/6, \ \pi/3, \ 2\pi/3, \ 4\pi/3, \ and \ 5\pi/3,\)
\(\displaystyle which \ reduces \ our \ solutions \ to \ eight.\)
\(\displaystyle Or, \ if \ you \ prefer \ in \ degrees \ 0^{0} \ \le \ x \ \le \ 360^{0}, \ then\)
\(\displaystyle 30^0, \ 60^0, \ 120^0, \ 150^0, \ 210^0, \ 240^0, \ 300^0, \ and \ 330^0 \ are \ all \ solutions \ to \ the \ equation.\)
