$\displaystyle \ \frac{{\cot A-\tan A}}{{\cot A+\tan A}}$

$\displaystyle (\cot A-\tan A)\div (\cot A+\tan A)$

Change tan A  to sin A/cos A and cot A to cos A/ sin A for easier manipulation.

$\displaystyle (\frac{{\cos A}}{{\sin A}}-\frac{{\sin A}}{{\cos A}})\div (\frac{{\cos A}}{{\sin A}}+\frac{{\sin A}}{{\cos A}})$

Find common denominators and simplify the fractions.

$\displaystyle (\frac{{\cos A\times \cos A}}{{\sin A\times \cos A}}-\frac{{\sin A\times \sin A}}{{\cos A\times \sin A}})\div (\frac{{\cos A\times \cos A}}{{\sin A\times \cos A}}+\frac{{\sin A\times \sin A}}{{\cos A\times \sin A}})$

$\displaystyle (\frac{{{{{\cos }}^{2}}A}}{{\sin A\cos A}}-\frac{{{{{\sin }}^{2}}A}}{{\cos A\sin A}})\div (\frac{{{{{\cos }}^{2}}A}}{{\sin A\cos A}}+\frac{{{{{\sin }}^{2}}A}}{{\cos A\sin A}})$

$\displaystyle (\frac{{{{{\cos }}^{2}}A-{{{\sin }}^{2}}A}}{{\sin A\cos A}})\div (\frac{{{{{\cos }}^{2}}A+{{{\sin }}^{2}}A}}{{\sin A\cos A}})$

Use the trigonometric identity and change cos2x + sin2x = 1.

$\displaystyle \frac{{{{{\cos }}^{2}}A-{{{\sin }}^{2}}A}}{{\sin A\cos A}}\div \frac{1}{{\sin A\cos A}}$

$\displaystyle \frac{{{{{\cos }}^{2}}A-{{{\sin }}^{2}}A}}{{\sin A\cos A}}\times \frac{{\sin A\cos A}}{1}$

$\displaystyle {{\cos }^{2}}A-{{\sin }^{2}}A$

$\displaystyle \cos 2A$

$\displaystyle 2{{\cos }^{2}}A-1$

Prove Trigonometric Identities involving Cotangent and Tangent . Additional Math Tuition. Woodlands, Choa Chu Kang, Yew Tee, Sembawang and Johor Bahru.