Square of Tangent Minus Square of Sine

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Theorem

$\tan^2 x - \sin^2 x = \tan^2 x \ \sin^2 x$

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \tan^2 x - \sin^2 x$	$=$	$\displaystyle \frac {\sin^2 x} {\cos^2x} - \sin^2 x$	$\displaystyle $	$\displaystyle $	$\displaystyle $	by definition of tangent
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac {\sin^2 x - \sin^2 x \ \cos^2 x} {\cos^2x}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac{\sin^2 x \left({1 - \cos^2 x}\right)} {\cos^2 x}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \tan^2 x \left({1 - \cos^2 x}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	by definition of tangent
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \tan^2 x \ \sin^2 x$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Sum of Squares of Sine and Cosine

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \tan^2 x - \sin^2 x\)	\(=\)	\(\displaystyle \frac {\sin^2 x} {\cos^2x} - \sin^2 x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of tangent
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {\sin^2 x - \sin^2 x \ \cos^2 x} {\cos^2x}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac{\sin^2 x \left({1 - \cos^2 x}\right)} {\cos^2 x}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \tan^2 x \left({1 - \cos^2 x}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of tangent
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \tan^2 x \ \sin^2 x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Sum of Squares of Sine and Cosine