${s i n}^{3} x + {c o s}^{3} x = (s i n x + c o s x) (1 - s i n x c o s x)$

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Solution

$s i n^{3} x + c o s^{3} x = (s i n x + c o s x) (1 - s i n x c o s x)$

$\frac{s i n^{3} x + c o s^{3} x}{s i n x + c o s x} = (1 - s i n x c o s x)$

LHS

$a^{3} + b^{3} = (a + b) (a^{2} - 2 a b + b^{2})$

$\frac{s i n^{3} x + c o s^{3} x}{s i n x + c o s x} = \frac{(s i n x + c o s x) (s i n^{2} x - s i n x c o s x + c o s^{2} x)}{s i n x + c o s x}$

$s i n^{2} x - s i n x c o s x + c o s^{2} x$

identity

$s i n^{2} x + c o s^{2} x = 1$

$s i n^{2} x - s i n x c o s x + c o s^{2} x$

$1 - s i n x c o s x$

hence,RHS is proved.

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