Question

What is the value of $\tan \left( 9 \right)-\tan \left( 27 \right)-\tan \left( 63 \right)+\tan \left( 81 \right)$ ?(all in degrees)

Answer 1

We know that we have studied six trigonometric functions till now and we need to use only one trigonometric function which is tangent here and have to evaluate it at various angles in order to find the value.

Complete step-by-step answer:
In the given question we are to find the values of the trigonometric function tangent at different angles. Also, these all-other functions are not involved so the calculation for the same would be easy and we will be able to find the values of tan angles from the trigonometric section.
$\begin{aligned} & \tan 81+\tan 9-\left( \tan 63+\tan 27 \right) \\\ & = \cot 9+\tan 9-\left( \cot 27+\tan 27 \right) \\\ & = \dfrac{{{\cos }^{2}}9+{{\sin }^{2}}9}{\cos 9\sin 9}-\dfrac{{{\cos }^{2}}27+{{\sin }^{2}}27}{\cos 27\sin 27} \\\ \end{aligned}$
Now, using the identity that ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$
We get,
$\begin{aligned} & \dfrac{1}{\cos 9\sin 9}-\dfrac{1}{\cos 27\sin 27} \\\ & = \dfrac{2}{\sin 18}-\dfrac{2}{\sin 54} \\\ & = 2\dfrac{\sin 54-\sin 18}{\sin 54\sin 18} \\\ \end{aligned}$
Now, further we get
$\begin{aligned} & = \dfrac{4\cos 36\sin 18}{\cos 36\sin 18} \\\ & = 4 \\\ \end{aligned}$
So, now the value of $\tan \left( 9 \right)-\tan \left( 27 \right)-\tan \left( 63 \right)+\tan \left( 81 \right)$ is going to be equal to 4.
Hence, the required answer to the given question is 4.

Note: We must take care to find the value of tangent function at different angles we don’t need to use the inverse of trigonometric functions which we sometimes use in order to get the answer but in this it will lead to wrong answers. Also, as the angles are in degrees so we don’t need to convert them into radian form and hence again convert them into degrees as in the last we need to answer the question in degrees and angles provided are also in degrees.