Show that $sinh (3 x) = 3 sinh x + 4 sin h^{3} x$

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Solution

LHS $= sinh (3 x)$
$sinh (3 x) = sinh (2 x + x)$
$= sinh (2 x) \times cosh (x) + cosh (2 x) \times sinh (x)$ [standardidentity]
$= 2 \times sinh (x) \times cosh (x) \times cosh (x) + (cos h^{2} (x) + sin h^{2} (x)) sinh (x)$
$= 2 \times sinh (x) \times cos h^{2} (x) + (1 + sin h^{2} (x) + sin h^{2} (x)) sinh (x) c o s h^{2} (x)$
$= 1 + sin h^{2} (x) = 2 sinh (x) (1 + sin h^{2} (x)) + (1 + 2 sin h^{2} (x)) sinh (x)$
$= 2 sinh (x) + 2 sin h^{3} (x) + sinh (x) + 2 sin h^{3} (x)$
$= 3 sinh (x) + 4 sin h^{3} (x)$
RHS proved.

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