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Relation between Inverses of Trigonometric Functions and Their Reciprocal Functions

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Question

Find the period of $S i n^{3} x + S i n^{2} x + 1 - c o s 2 x$

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Solution

$f (x) = {sin}^{3} x + {sin}^{2} x + 1 - {cos}^{2} x$
$= {sin}^{3} x + {sin}^{2} x + 2 {sin}^{2} x$
$= {sin}^{3} x + 3 {sin}^{2} x$
Period of ${sin}^{3} x = 2 π$
Period of ${sin}^{2} x = π$
$\Rightarrow$ Period of $f (x) = L . C . M (π, 2 π)$
$= 2 π$

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