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For any $\theta \, \in \,\left( {\frac{\pi }{4},\frac{\pi }{2}} \right)$, the expression $3\,{\left( {\sin \,\theta - \cos \,\theta } \right)^4} + 6{\left( {\sin \,\theta + \cos \,\theta } \right)^2} + 4\,{\sin ^6}\,\theta $ equals
$13 - 4\,{\cos ^2}\,\theta \, + 6\,{\sin ^2}\,\theta \,{\cos ^2}\,\theta $
$13 - 4\,{\cos ^6}\,\theta \,$
$13 - 4\,{\cos ^2}\,\theta \, + 6\,\,{\cos ^4}\,\theta $
$13 - 4\,{\cos ^4}\,\theta \, + 2\,{\sin ^2}\,\theta \,{\cos ^2}\,\theta $
Solution
$3\,{(1 – \sin 2\theta )^2}\, + \,6(1 + \sin 2\theta )\, + \,4\,{\sin ^6}\theta $
$ = 3\,(1 – 2\sin \,2\theta + {\sin ^2}2\theta ) + \,6 + 6\sin 2\theta + \,4\,{\sin ^6}\theta $
$ = \,9 + 3{\sin ^2}2\theta + 4\,{\sin ^6}\theta $
$ = \,9 + 12{\sin ^2}\theta {\cos ^2}\theta + 4\,{(1 – {\cos ^2}\theta )^3}$
$ = \,9 + 12(1 – {\cos ^2}\theta ){\cos ^2}\theta + 4\,(1 – 3{\cos ^2}\theta + 3{\cos ^4}\theta – {\cos ^6}\theta )$
$ = \,13 + 12{\cos ^2}\theta – 12{\cos ^4}\theta – 12{\cos ^2}\theta + 12{\cos ^4}\theta – 4{\cos ^6}\theta $
$ = \,13 – 4{\cos ^6}\theta $
