Question

How do you write the expression as the sine, cosine, or the tangent of the angle given \[\dfrac{\tan {{68}^{\circ }}-\tan {{115}^{\circ }}}{1+\tan {{68}^{\circ }}\tan {{115}^{\circ }}}\]?

Answer 1

Hint: To solve the given question, we need to know the trigonometric expansions of the difference of angles formula for the ratio tangent. The expansion formula for tangent is \[\tan \left( a-b \right)=\dfrac{\tan a-\tan b}{1+\tan a\times \tan b}\]. We will use this formula to simplify the given expression and express it in the tangent of an angle. To do this, we first need to find the values of the variables a and b for the given expression.

Complete step by step answer:
We are given the trigonometric expression as \[\dfrac{\tan {{68}^{\circ }}-\tan {{115}^{\circ }}}{1+\tan {{68}^{\circ }}\tan {{115}^{\circ }}}\]. This expression is similar to the expansion formula for difference of angles of tangent \[\dfrac{\tan a-\tan b}{1+\tan a\times \tan b}\] on simplifying we can write this as \[\tan \left( a-b \right)\]. Comparing this formula, we get \[a={{68}^{\circ }}\And b={{115}^{\circ }}\]. Using the expansion formula, we can write the given expression as
\[\begin{align}
  & \Rightarrow \dfrac{\tan {{68}^{\circ }}-\tan {{115}^{\circ }}}{1+\tan {{68}^{\circ }}\tan {{115}^{\circ }}} \\
 & \Rightarrow \tan \left( {{68}^{\circ }}-{{115}^{\circ }} \right) \\
\end{align}\]
Subtracting 115 from 68, we get \[-47\], substituting this value in the above expression. We get
\[\Rightarrow \tan \left( -{{47}^{\circ }} \right)\]
Using the property which states \[\tan \left( -x \right)=-\tan x\]. Here, the value of x is 47 degrees. We can simplify the expression using the property as
\[\Rightarrow \tan \left( -{{47}^{\circ }} \right)=-\tan {{47}^{\circ }}\]

Hence, the trigonometric expression \[\dfrac{\tan {{68}^{\circ }}-\tan {{115}^{\circ }}}{1+\tan {{68}^{\circ }}\tan {{115}^{\circ }}}\] can be expressed as \[-\tan {{47}^{\circ }}\] on simplifying.

Note: To solve these types of questions, one should know the different trigonometric expansion formulas. For example, sum of angle, difference of angles. The difference of angles formula for another trigonometric ratio can be expressed as follows:
For sine ratio \[\sin \left( a-b \right)=\sin a\cos b-\cos a\sin b\], for cosine ratio
These formulas can be used for angles in degree as well as angles in radian measure also.