Question

If $\tan \varphi + \cot \varphi = 2$ then the value of ${\tan ^{100}}\varphi + {\cot ^{100}}\varphi = $
A) 1
B) 2
C) 3
D) 0

Answer 1

Hint:
From the given condition we use the ratio $\cot \varphi = \dfrac{{\sin \varphi }}{{\cos \varphi }}$and hence we get a quadratic equation in ${\tan ^2}\varphi $ and solving this using splitting the middle term method we get the value of $\varphi $and using this we get the value of ${\tan ^{100}}\varphi + {\cot ^{100}}\varphi $

Complete step by step solution:
We are given that $\tan \varphi + \cot \varphi = 2$………..(1)
We know that $\cot \varphi = \dfrac{{\sin \varphi }}{{\cos \varphi }}$
This is nothing other than $\dfrac{1}{{\dfrac{{\sin \varphi }}{{\cos \varphi }}}} = \dfrac{1}{{\tan \varphi }}$
Using this in (1) we get
$
   \Rightarrow \tan \varphi + \dfrac{1}{{\tan \varphi }} = 2 \\
   \Rightarrow {\tan ^2}\varphi + 1 = 2\tan \varphi \\
   \Rightarrow {\tan ^2}\varphi - 2\tan \varphi + 1 = 0 \\
 $
Now we get a quadratic equation in ${\tan ^2}\varphi $
This can be solved using splitting the middle term method
Hence here our middle term can be split as $ - \tan \varphi - \tan \varphi $
$
   \Rightarrow {\tan ^2}\varphi - \tan \varphi - \tan \varphi + 1 = 0 \\
   \Rightarrow \tan \varphi \left( {\tan \varphi - 1} \right) - 1\left( {\tan \varphi - 1} \right) = 0 \\
   \Rightarrow \left( {\tan \varphi - 1} \right)\left( {\tan \varphi - 1} \right) = 0 \\
   \Rightarrow {\left( {\tan \varphi - 1} \right)^2} = 0 \\
 $
Hence here we get that $\tan \varphi = 1$
Using the trigonometric values , we know that $\tan 45 = 1$
Hence $\varphi = 45$ ……(2)
Now we are asked to find the value of ${\tan ^{100}}\varphi + {\cot ^{100}}\varphi $
Lets use the value of $\varphi $ in the above equation
$
   \Rightarrow {\tan ^{100}}45 + {\cot ^{100}}45 \\
   \Rightarrow {\left( {\tan 45} \right)^{100}} + {\left( {\cot 45} \right)^{100}} \\
   \Rightarrow {\left( 1 \right)^{100}} + {\left( 1 \right)^{100}} \\
   \Rightarrow 1 + 1 \\
   \Rightarrow 2 \\
 $
Hence we get the required value

Therefore the correct option is B.

Additional information :
1) The other trigonometric values of tan are given as
$\tan 0 = 0;\tan 30 = \dfrac{1}{{\sqrt 3 }};\tan 60 = \sqrt 3 $
2) Steps to keep in mind while solving trigonometric problems are
2.1) Always start from the more complex side
2.2) Express everything into sine and cosine

Note:
Here the quadratic equation can also be solved using the quadratic formula $\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
$ \Rightarrow {\tan ^2}\varphi - 2\tan \varphi + 1 = 0$
Here a = 1 , b = -2 and c = 1
$
   \Rightarrow \tan \varphi = \dfrac{{2 \pm \sqrt {{{( - 2)}^2} - 4(1)(1)} }}{{2(1)}} \\
   \Rightarrow \tan \varphi = \dfrac{{2 \pm \sqrt {4 - 4} }}{2} \\
   \Rightarrow \tan \varphi = \dfrac{2}{2} = 1 \\
 $