Question

How do you simplify the expression, $\sin \left( \theta \right)+\cos \left( \theta \right)\tan \left( \theta \right)$?

Answer 1

Hint: Now we are given with a trigonometric expression. To simplify the expression we will use the formula $\tan \theta =\dfrac{\sin \theta }{\cos \theta }$ and substitute the value of tan in the given expression. Now we will simplify the given expression by cancelling the common terms in numerator and denominator. Hence we will get a simplified expression.

Complete step by step solution:
Now we are given a trigonometric expression where sin, cos and tan are the trigonometric ratios.
Now let us first understand the functions sin, cos and tan.
Now in a right angle triangle if $\theta $ is an angle then the trigonometric ratios are defined as.
$\sin \left( \theta \right)=\dfrac{\text{opposite side}}{\text{hypotenuse}}$ , $\cos \left( \theta \right)=\dfrac{\text{adjacent side}}{\text{hypotenuse}}$ and $\tan \left( \theta \right)=\dfrac{\text{opposite side}}{\text{adjacent side}}$ .
Now let us take the ratio of sin and cos.
\[\Rightarrow \dfrac{\sin \theta }{\cos \theta }=\dfrac{\dfrac{\text{opposite side}}{\text{hypotenuse}}}{\dfrac{\text{adjacent side}}{\text{hypotenuse}}}\]
Now on simplifying the fraction on RHS we get,
\[\Rightarrow \dfrac{\sin \theta }{\cos \theta }=\dfrac{\text{opposite side}}{\text{adjacent side}}\]
Now we know that $\tan \left( \theta \right)=\dfrac{\text{opposite side}}{\text{adjacent side}}$
Hence we can say that $\dfrac{\sin \theta }{\cos \theta }=\tan \theta $
Now consider the given equation $\sin \theta +\cos \theta \left( \tan \theta \right)$ .
Let us substitute the value of $\tan \theta $ Hence we get,
$\Rightarrow \sin \theta +\cos \theta \left( \tan \theta \right)=\sin \theta +\cos \theta \left( \dfrac{\sin \theta }{\cos \theta } \right)$
Now on simplifying the equation by cancelling cos in numerator and denominator we get,
$\begin{align}
  & \Rightarrow \sin \theta +\cos \theta \left( \tan \theta \right)=\sin \theta +\sin \theta \\
 & \Rightarrow \sin \theta +\cos \theta \left( \tan \theta \right)=2\sin \theta \\
\end{align}$
Hence after simplification the given equation becomes $2\sin \theta $ .

Note: Now note that thought tan is just the ratio of sin and cos the functions are very different. For example since tan has cos in the denominator. Hence the function is not defined for the values when cosx = 0. While for sin and cos we have the function is defined for all values of x. Also for sin and cos the range of the function is [-1,1] while for tan the range is the whole set of real numbers.