tan(x) from sin(x) and cos(x)

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Given sin(x) and cos(x) can I determine the value of tan(x) without using tan itself (i.e: no sin^-1(sin(x)) etc)?

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Yes: $$ \tan x = \frac{\sin x}{\cos x} $$

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We have $\tan x=\frac{\sin x}{\cos x}$.