Question

How do you simplify \[\tan \theta + \cos \left( { - \theta } \right) + \tan \left( { - \theta } \right)\]?

Answer 1

Hint: To simplify the given question, we will use the trigonometric properties of cosines and tangent function. We will first make all the terms containing the negative angles change into terms containing positive angles and then simplify to get the required answer.

Complete step by step answer:
We have to simplify \[\tan \theta + \cos \left( { - \theta } \right) + \tan \left( { - \theta } \right)\].
We know that \[\cos \left( { - \theta } \right) = \cos \theta \], and also \[\tan \left( { - \theta } \right) = - \tan \theta \].
So, using this we will replace the terms with negative angles. So, we have;
\[ = \tan \theta + \cos \theta - \tan \theta \]
On simplification we get;
\[ = \cos \theta \]

Note:
One mistake that students may commit is that on seeing the question they might immediately think of converting the whole question in terms of sine and cosine terms. But here there is no need for that. we can even directly solve the question just by changing the angles.