If the trigonometric equation is given as \[\tan {{20}^{\circ }}\tan {{40}^{\circ }}\tan {{60}^{\circ }}\tan {{80}^{\circ }}=k\], then find the value of k.

Hint: Simplify the given trigonometric expression using the trigonometric properties of tangent function. Also use the trigonometric identity \[\tan x\tan \left( {{60}^{\circ }}-x \right)\tan \left( {{60}^{\circ }}+x \right)=\tan 3x\] to simplify the given trigonometric expression.

Complete step-by-step solution -
We have the trigonometric expression \[\tan {{20}^{\circ }}\tan {{40}^{\circ }}\tan {{60}^{\circ }}\tan {{80}^{\circ }}=k\]. We have to calculate the value of the variable ‘k’.
We know the trigonometric identity \[\tan x\tan \left( {{60}^{\circ }}-x \right)\tan \left( {{60}^{\circ }}+x \right)=\tan 3x\].
Substituting \[x={{20}^{\circ }}\] in the above identity, we have \[\tan {{20}^{\circ }}\tan {{40}^{\circ }}\tan {{80}^{\circ }}=\tan \left( 3\times {{20}^{\circ }} \right)=\tan \left( {{60}^{\circ }} \right)\].
Thus, we can write \[\tan {{20}^{\circ }}\tan {{40}^{\circ }}\tan {{60}^{\circ }}\tan {{80}^{\circ }}=k\] as \[{{\tan }^{2}}{{60}^{\circ }}=k\].
We know that \[\tan {{60}^{\circ }}=\sqrt{3}\].
Thus, we have \[k={{\left( \sqrt{3} \right)}^{2}}=3\].
Hence, on simplifying the expression \[\tan {{20}^{\circ }}\tan {{40}^{\circ }}\tan {{60}^{\circ }}\tan {{80}^{\circ }}=k\], the value of k is 3.
Trigonometric functions are real functions which relate any angle of a right angled triangle to the ratios of any two of its sides. The widely used trigonometric functions are sine, cosine and tangent. However, we can also use their reciprocals, i.e., cosecant, secant and cotangent. We can use geometric definitions to express the value of these functions on various angles using unit circle (circle with radius 1). We also write these trigonometric functions as infinite series or as solutions to differential equations. Thus, allowing us to expand the domain of these functions from the real line to the complex plane.

Note: One should be careful while using the trigonometric identities and rearranging the terms to convert from one trigonometric function to the other one. We must also keep in mind that all the angles given to us are in degrees, not in radians. If we consider them in radians, we will get an incorrect answer. One must know the value of tangent function at certain angles; otherwise, we won’t be able to simplify the given trigonometric expression and find the value of ‘k’.