Question

Find the value of \[\tan \left( {{225}^{\circ }} \right)+\cot \left( {{135}^{\circ }} \right)\].

Answer 1

Hint: In order to find the value of \[\tan \left( {{225}^{\circ }} \right)+\cot \left( {{135}^{\circ }} \right)\], firstly we will be finding the nearest principle angle to the angle given and then we will be expressing the given value of angles in terms of principle angle. And then upon substituting and solving the angles, we obtain the required value of the given functions.

Complete step-by-step solution:
Let us have a brief regarding the trigonometric functions. The counter-clockwise angle between the initial arm and the terminal arm of an angle in standard position is called the principal angle. Its value is between \[{{0}^{\circ }}\] and \[{{360}^{\circ }}\]. The relationship between the angles and sides of a triangle are given by the trigonometric functions. The basic trigonometric functions are sine, cosine, tangent, cotangent, secant and cosecant. These are the basic main trigonometric functions used.
Now let us find the value of \[\tan \left( {{225}^{\circ }} \right)+\cot \left( {{135}^{\circ }} \right)\].
Using the principle angles, we can write the given trigonometric functions as;
\[\begin{align}
  & \tan \left( {{225}^{\circ }} \right)=\tan \left( {{270}^{\circ }}-{{45}^{\circ }} \right) \\
 & \cot \left( {{135}^{\circ }} \right)=\cot \left( {{180}^{\circ }}-{{45}^{\circ }} \right) \\
\end{align}\]
Now let us substitute them and solve it.
\[\Rightarrow \tan \left( {{270}^{\circ }}-{{45}^{\circ }} \right)+\cot \left( {{180}^{\circ }}-{{45}^{\circ }} \right)\]
Upon solving this, we get
\[\begin{align}
  & \Rightarrow \tan \left( {{270}^{\circ }}-{{45}^{\circ }} \right)+\cot \left( {{180}^{\circ }}-{{45}^{\circ }} \right) \\
 & \Rightarrow \tan \left( 3{{\left( 90 \right)}^{\circ }}-{{45}^{\circ }} \right)+\cot \left( 2\left( {{90}^{\circ }} \right)-{{45}^{\circ }} \right) \\
 & \Rightarrow \cot {{45}^{\circ }}-\cot {{45}^{\circ }} \\
 & =0 \\
\end{align}\]
\[\therefore \] The value of \[\tan \left( {{225}^{\circ }} \right)+\cot \left( {{135}^{\circ }} \right)\] is \[0\].

Note: We must be aware of the operations on trigonometric angles being done. In above problem, we can notice that \[\tan \] changes to \[\cot \] because \[\tan \left( {{90}^{\circ }}-\theta \right)=-\cot \theta \]. The common error could be improper conversion of trigonometric functions and not considering the changes in the sign of the function. Sometimes, according to our convenience we can convert the degrees into radians and radians into degrees as per requirement for easy solving of the problems.