Chapter 7.3, Problem 10MCQ

To determine

To find: the exact value of $\text{tan}\left(x-y\right)$ by using the given information.

Answer to Problem 10MCQ

  $\frac{21\sqrt{3}-20}{59}$

Explanation of Solution

Given information:

  $\begin{array}{l}\tan x=\frac{5}{4},0<x<\frac{\pi }{2}\\ \sec y=2,0<y<\frac{\pi }{2}\end{array}$

Calculation:

Evaluate the value of $\tan y$ as shown below.

  $\begin{array}{c}\tan y=\sqrt{{\sec }^{2}y-1}\left[\text{Pythagorean identity}\right]\\ =\sqrt{{\left(2\right)}^2-1}\left[\text{Put the value of }\sec y\right]\\ =\sqrt{4-1}\\ =\sqrt{3}\end{array}$

Now solve $\text{tan}\left(x-y\right)$ by applying difference identity for tangent as shown below.

  $\begin{array}{c}\tan \left(x-y\right)=\frac{\tan x-\tan y}{1+\tan x\cdot \tan y}\\ =\frac{\left(\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\right)-\left(\sqrt{3}\right)}{1+\left(\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\right)\cdot \left(\sqrt{3}\right)}\left[\text{Put the values for tangent functions}\right]\\ =\frac{\left(\frac{5-4\sqrt{3}}{4\sqrt{3}}\right)}{\left(\frac{4+5\sqrt{3}}{4\sqrt{3}}\right)}\\ =\frac{5-4\sqrt{3}}{4+5\sqrt{3}}\\ =\frac{5-4\sqrt{3}}{4+5\sqrt{3}}\left(\frac{4-5\sqrt{3}}{4-5\sqrt{3}}\right)\left[\text{Multiply and divide by }\left(4-5\sqrt{3}\right)\right]\\ =\frac{20-16\sqrt{3}-25\sqrt{3}+20\sqrt{3}}{16-75}\\ =\frac{20-21\sqrt{3}}{-59}\\ =\frac{21\sqrt{3}-20}{59}\end{array}$

Thus the value of given trigonometric function is $\frac{21\sqrt{3}-20}{59}$ .