Evaluate the following: Show that: 1- 60^ 60^ = 60^ -1 60^ +1

Question

Evaluate the following:
Show that:
$\frac{1-\sin60^\circ}{\cos60^\circ}=\frac{\tan60^\circ-1}{\tan60^\circ+1}$

✓

Answer

$\text{L.H.S.}=\frac{1-\sin60^\circ}{\cos60^\circ}=\frac{\frac{\sqrt{3}}{2}}{\frac12}$
$=\frac{\Big(\frac{2-\sqrt{3}}{2}\Big)}{\frac{1}{2}}=\Big(\frac{2-\sqrt{3}}{2}\Big)\times2=2-\sqrt{3}$
$\text{R.H.S.}=\frac{\tan60^\circ-1}{\tan60^\circ+1}=\frac{\sqrt{3}-1}{\sqrt{3}+1}$
$=\frac{\sqrt{3}-1}{\sqrt{3}+1}\times\frac{\sqrt{3}-1}{\sqrt{3}-1}=\frac{\big(\sqrt{3}-1\big)^2}{\big(\sqrt{3}\big)^2-1^2}$
$=\frac{3+1-2\sqrt{3}}{3-1}=\frac{4-2\sqrt{3}}{2}=2-\sqrt{3}$
Hence, L.H.S. = R.H.S.

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