Question

Find the value of \[tan\left( { - {{135}^ \circ }} \right)\] and \[\cot \left( { - {{135}^ \circ }} \right)\] along with the steps.

Answer 1

Hint:Using the property that the reciprocal of a tangent function is nothing but a cotangent function itself. As, we know the trigonometry identity that \[\tan ( - x) = - \tan x\]. We will also use the trigonometry identity \[\tan x = \dfrac{{\sin x}}{{\cos x}}\]. We also know the inverse of \[\tan x = \dfrac{1}{{\cot x}}\]. And, we will use all the identities to find the value of tan(x) and then we will use the inverse of tan(x) to find the value of cot(x).

Complete step by step answer:
We know the trigonometry identity \[\tan x = \dfrac{{\sin x}}{{\cos x}}\].
We will use this identity for x = -x as,
\[\tan ( - x) = \dfrac{{\sin ( - x)}}{{\cos ( - x)}}\]
As, we know that, \[\sin ( - x) = - \sin (x)\]and \[\cos ( - x) = \cos (x)\]
\[\tan ( - x) = \dfrac{{ - \sin (x)}}{{\cos (x)}} \\
\Rightarrow \tan ( - x) = - \tan (x) \]
Thus, we use this identity to solve and get the required value.

Let, apply the trigonometry identity for\[\tan ( - {135^ \circ })\], we get,
\[\tan ( - {135^ \circ }) = - \tan ({135^ \circ })\]
Now we will split\[135 = 90 + 45\], we will get,
\[ \Rightarrow \tan ( - {135^ \circ }) = - \tan ({90^ \circ } + {45^ \circ })\]
We will apply the trigonometry identity that\[\tan ({90^ \circ } + x) = - \cot x\]
Here, \[x = {45^ \circ }\]then we will get,
\[ \Rightarrow \tan ( - {135^ \circ }) = - ( - \cot {45^ \circ })\]
\[ \Rightarrow \tan ( - {135^ \circ }) = \cot {45^ \circ }\]
We know that, \[\cot {45^ \circ } = 1\].
\[ \Rightarrow \tan ( - {135^ \circ }) = 1\]
Thus, the value of \[\tan ( - {135^ \circ }) = 1\].

Next, we will find the value of\[\cot ( - {135^ \circ })\] as below,
According to the trigonometry identity,
\[\tan ( - x) = \dfrac{1}{{\cot ( - x)}}\]
\[ \Rightarrow \cot ( - x) = \dfrac{1}{{\tan ( - x)}}\]
Let, \[x = {135^ \circ }\]
Then,
\[\cot ( - {135^ \circ }) = \dfrac{1}{{\tan ( - {{135}^ \circ })}}\]
\[ \Rightarrow \cot ( - {135^ \circ }) = \dfrac{1}{1}\]
\[ \Rightarrow \cot ( - {135^ \circ }) = 1\]
Thus, the value of \[\cot ( - {135^ \circ }) = 1\].

Hence, the value of \[\tan ( - {135^ \circ }) = 1\] and \[\cot ( - {135^ \circ }) = 1\].

Note: (1) Another method to solve,
\[\tan ( - {135^ \circ }) = \tan ({360^ \circ } - {135^ \circ }) \\
\Rightarrow \tan ( - {135^ \circ }) = \tan ({225^ \circ }) \]
Now,
\[\tan ({225^ \circ }) = \tan ({180^ \circ }) + \tan ({45^ \circ })\]
\[\Rightarrow \tan ({225^ \circ }) = \dfrac{{\tan ({{180}^ \circ }) + \tan ({{45}^ \circ })}}{{1 - \tan ({{180}^ \circ })\tan ({{45}^ \circ })}}\]
We know that \[\tan {45^ \circ } = 1\] and \[\tan {180^ \circ } = 0\], we will apply this and we will get,
\[\tan ( - {135^ \circ }) = \dfrac{{0 + 1}}{{1 + 0}}\]
\[\therefore \tan ( - {135^ \circ })= \dfrac{1}{1} = 1 \]
Thus, the value of \[\tan ( - {135^ \circ }) = 1\].

(2) Another Method: If we split 135 = 180-45, we
\[ \Rightarrow \tan ( - {135^ \circ }) = - \tan ({180^ \circ } - {45^ \circ })\]
\[\Rightarrow \tan ( - {135^ \circ }) = - \tan ({135^ \circ })\]
\[\Rightarrow \tan ( - {135^ \circ })= - \tan ({180^ \circ } - {45^ \circ })\]
As, we know that\[\tan ({180^ \circ } - x) = - \tan x\].
Here, \[x = {45^ \circ }\] and apply this, we will get,
\[\Rightarrow \tan ( - {135^ \circ }) = - ( - \tan {45^ \circ })\]
\[\Rightarrow \tan ( - {135^ \circ }) = - ( - 1)\]....................\[(\because \tan {45^ \circ } = 1)\]
\[ \therefore \tan ( - {135^ \circ })= 1\]
Thus, the value of \[\tan ( - {135^ \circ }) = 1\].