Question

What is the value of \[\dfrac{\cos {{9}^{\circ }}+\sin {{9}^{\circ }}}{\cos {{9}^{\circ }}-\sin {{9}^{\circ }}}\] equal to
$1)\text{ }\tan {{26}^{\circ }}$
$2)\text{ }\tan {{81}^{\circ }}$
$3)\text{ }\tan {{51}^{\circ }}$
$4)\text{ }\tan {{54}^{\circ }}$
$5)\text{ }\tan {{46}^{\circ }}$

Answer 1

Hint: In this question we have the expression given to us as the division of trigonometric functions. We will solve this question by dividing both the numerator and denominator by $\cos {{9}^{\circ }}$ so that we get the expression in the form of $\tan $. We will then use the formula that $\tan {{45}^{\circ }}=1$ and then use the identity $\dfrac{\tan A+\tan B}{1-\tan A\tan B}=\tan \left( A+B \right)$ to get the required solution.

Complete step-by-step solution:
We have the expression given to us as:
\[\Rightarrow \dfrac{\cos {{9}^{\circ }}+\sin {{9}^{\circ }}}{\cos {{9}^{\circ }}-\sin {{9}^{\circ }}}\]
On dividing both the numerator and denominator by $\cos {{9}^{\circ }}$, we get:
\[\Rightarrow \dfrac{\dfrac{\cos {{9}^{\circ }}+\sin {{9}^{\circ }}}{\cos {{9}^{\circ }}}}{\dfrac{\cos {{9}^{\circ }}-\sin {{9}^{\circ }}}{\cos {{9}^{\circ }}}}\]
On splitting the denominator, we get:
\[\Rightarrow \dfrac{\dfrac{\cos {{9}^{{}^\circ }}}{\cos {{9}^{{}^\circ }}}+\dfrac{\sin {{9}^{{}^\circ }}}{\cos {{9}^{{}^\circ }}}}{\dfrac{\cos {{9}^{{}^\circ }}}{\cos {{9}^{{}^\circ }}}-\dfrac{\sin {{9}^{{}^\circ }}}{\cos {{9}^{{}^\circ }}}}\]
Now we know that $\dfrac{\sin a}{\cos a}=\tan a$ therefore, on substituting and simplifying, we get:
\[\Rightarrow \dfrac{1+\tan {{9}^{{}^\circ }}}{1-\tan {{9}^{{}^\circ }}}\]
Now we can rewrite the expression as:
\[\Rightarrow \dfrac{1+\tan {{9}^{{}^\circ }}}{1-1\times \tan {{9}^{{}^\circ }}}\]
Now we know that $\tan {{45}^{\circ }}=1$ therefore, on substituting it, we get:
\[\Rightarrow \dfrac{\tan {{45}^{\circ }}+\tan {{9}^{{}^\circ }}}{1-\tan {{45}^{\circ }}\times \tan {{9}^{{}^\circ }}}\]
Now we know the formula that $\dfrac{\tan A+\tan B}{1-\tan A\tan B}=\tan \left( A+B \right)$, and the above expression is in the form of the given formula therefore, we can write the expression as:
\[\Rightarrow \tan \left( {{45}^{\circ }}+{{9}^{\circ }} \right)\]
On adding the terms, we get:
\[\Rightarrow \tan {{54}^{\circ }}\], which is the required solution.
Therefore, the correct option is $\left( 4 \right)$.

Note: It is to be remembered that whenever the value of the angle is given in the expression it should be expanded and simplified such that it yields a value for which the value is known. This makes the expression more simplified when the value is substituted. In this question we have used the trigonometric addition-subtraction formula for the angles.