Question

Prove that following identities:

$tan\theta tan(\theta +{60}^{\circ })+tan\theta tan(\theta -{60}^{\circ })+tan(\theta +{60}^{\circ })tan(\theta -{60}^{\circ })=-3$

Answer 1

$tanA\times tan(A+{60}^{\circ })+tanA\times tan(A-{60}^{\circ })+tan(A+{60}^{\circ })tan(A-{60}^{\circ })=tan\left(A\right)\frac{\left[tan\right(A)-tan({60}^{\circ }\left)\right]}{[1+tan(A\left)tan\right({60}^{\circ }\left)\right]}+tan\left(A\right)\frac{\left[tan\right(A)+tan({60}^{\circ }\left)\right]}{[1-tan(A\left)tan\right({60}^{\circ }\left)\right]}+\left\{\frac{\left[tan\right(A)-tan({60}^{\circ }\left)\right]}{[1+tan(A\left)tan\right({60}^{\circ }\left)\right]}\right\}\left\{\frac{\left[tan\right(A)+tan({60}^{\circ }\left)\right]}{[1-tan(A\left)tan\right({60}^{\circ }\left)\right]}\right\}$

$=tan\left(A\right)\frac{\left[tan\right(A)-tan({60}^{\circ }\left)\right][1-tan(A\left)tan\right({60}^{\circ }\left)\right]}{[1-ta{n}^2(A\left)ta{n}^2\right({60}^{\circ }\left)\right]}+tan\left(A\right)\frac{\left[tan\right(A)+tan({60}^{\circ }\left)\right][1+tan(A\left)tan\right({60}^{\circ }\left)\right]}{[1-ta{n}^2(A\left)ta{n}^2\right({60}^{\circ }\left)\right]}+\frac{\left[tan\right(A)-tan({60}^{\circ }\left)\right]\left[tan\right(A)+tan({60}^{\circ }\left)\right]}{[1-ta{n}^2(A\left)ta{n}^2\right({60}^{\circ }\left)\right]}$

$=tan\left(A\right)\frac{\left[tan\right(A)-\sqrt{3}][1-\sqrt{3}tan(A\left)\right]}{[1-3ta{n}^2(A\left)\right]}+tan\left(A\right)\frac{\left[tan\right(A)+\sqrt{3}][1+\sqrt{3}tan(A\left)\right]}{[1-3ta{n}^2(A\left)\right]}+\frac{\left[tan\right(A)-\sqrt{3}]\left[tan\right(A)+\sqrt{3}]}{[1-3ta{n}^2(A\left)\right]}tan\left(A\right)\frac{\left[4tan\right(A)-\sqrt{3}-\sqrt{3}ta{n}^2(A\left)\right]}{[1-3ta{n}^2(A\left)\right]}+tan\left(A\right)\frac{4tan\left(A\right)+\sqrt{3}+\sqrt{3}ta{n}^2\left(A\right)}{[1-3ta{n}^2(A\left)\right]}+\frac{\left[ta{n}^2\right(A)-3]}{[1-3ta{n}^2(A\left)\right]}$

$=\frac{\left[9ta{n}^2\right(A)-3]}{[1-3ta{n}^2(A\left)\right]}=-3\frac{[1-3ta{n}^2(A\left)\right]}{[1-3ta{n}^2(A\left)\right]}$

=-3