Question

What is the value of$\dfrac{\cos {{10}^{\circ }}+\sin {{10}^{\circ }}}{\cos {{10}^{\circ }}-\sin {{10}^{\circ }}}$?
A. $\tan {{55}^{\circ }}$
B. $\cot {{55}^{\circ }}$
C. $-\tan {{35}^{\circ }}$
D. $-\cot {{35}^{\circ }}$

Answer 1

Hint: In this question, we have to find the value of $\dfrac{\cos {{10}^{\circ }}+\sin {{10}^{\circ }}}{\cos {{10}^{\circ }}-\sin {{10}^{\circ }}}$. As we see the options all are in tan and the cot form. So first we try to make the equation in tan form. For this, we divide the numerator and denominator by $\cos {{10}^{\circ }}$. Then we apply the formula of $\tan (x+y)$ and solving and comparing the values, we get the desirable answer and choose the correct option.

Formula Used:
In this question, we use the trigonometric formula which is :-
$\tan \theta =\dfrac{\sin \theta }{\cos \theta }$
 And $\tan (x+y)=\dfrac{\tan x+\tan y}{1-\tan x\tan y}$

Complete Step- by- step Solution:
We have to find the value of $\dfrac{\cos {{10}^{\circ }}+\sin {{10}^{\circ }}}{\cos {{10}^{\circ }}-\sin {{10}^{\circ }}}$
To find the value, first we divide the numerator and denominator of the given equation by $\cos {{10}^{\circ }}$
we get $\dfrac{\dfrac{\cos {{10}^{\circ }}}{\cos {{10}^{\circ }}}+\dfrac{\sin {{10}^{\circ }}}{\cos {{10}^{\circ }}}}{\dfrac{\cos {{10}^{\circ }}}{\cos {{10}^{\circ }}}-\dfrac{\sin {{10}^{\circ }}}{\cos {{10}^{\circ }}}}$…………………………………….. (1)
We know $\tan \theta =\dfrac{\sin \theta }{\cos \theta }$
By solving the equation (1), we get $\dfrac{1+\tan {{10}^{\circ }}}{1-\tan {{10}^{\circ }}}$
 We know the value of $\tan {{45}^{\circ }}=1$
 So $\dfrac{1+\tan {{10}^{\circ }}}{1-\tan {{10}^{\circ }}}$ = $\dfrac{\tan {{45}^{\circ }}+\tan {{10}^{\circ }}}{1-\tan {{45}^{\circ }}\tan {{10}^{\circ }}}$……………………… (2)
As we know the formula of $\tan (x+y)=\dfrac{\tan x+\tan y}{1-\tan x\tan y}$……………………………. (3)
Now by comparing the equation (2) with equation (3),
We get $\dfrac{\tan {{45}^{\circ }}+\tan {{10}^{\circ }}}{1-\tan {{45}^{\circ }}\tan {{10}^{\circ }}}$ = $\tan ({{45}^{\circ }}+{{10}^{\circ }})$
And $\tan ({{45}^{\circ }}+{{10}^{\circ }})$ = $\tan {{55}^{\circ }}$
Hence the value of $\dfrac{\cos {{10}^{\circ }}+\sin {{10}^{\circ }}}{\cos {{10}^{\circ }}-\sin {{10}^{\circ }}}$ = $\tan {{55}^{\circ }}$

Thus, Option (A) is the correct answer.

Note: In trigonometry, different type of questions can be solved with the help of different trigonometry formulas. These questions may include trigonometric ratios, Pythagoras theorem, product identities etc. There are many trigonometry formulas. We choose the formula according to our question demands. Learning and remembering these formulas help the student to solve the questions accurately and able to choose the correct option.