If $\tan \theta =\dfrac{3}{4}$ then ${{\cos }^{2}}\theta -{{\sin }^{2}}\theta $ is equal to
a)$\dfrac{7}{25}$
b)$1$
c)$\dfrac{-7}{25}$
d)$\dfrac{4}{25}$
VerifiedHint: Here, the opposite side and adjacent side can be identified by the formula $\tan \theta =\dfrac{opposite\text{ }side}{adjacent\text{ }side}$. Then, construct a right triangle. With the help of the right triangle, find the hypotenuse using Pythagoras theorem. Next, find the values of $\sin \theta $ and $\cos \theta $, substitute these values in the expression ${{\cos }^{2}}\theta -{{\sin }^{2}}\theta $.
Complete step-by-step answer:
Here, we are given that $\tan \theta =\dfrac{3}{4}$.
Now, we have to find the value of ${{\cos }^{2}}\theta -{{\sin }^{2}}\theta $.
Now, consider the figure,
We know that,
$\tan \theta =\dfrac{opposite\text{ }side}{adjacent\text{ }side}$
In the figure we have,
Opposite side = AC
Adjacent side = AB
Hypotenuse = BC
$\Delta ABC$ is a right angled triangle. Hence, we can apply the Pythagoras theorem.
Now, by Pythagoras theorem we have,
$ {{(Hypotenuse)}^{2}}={{(Opposite\text{ }side)}^{2}}+{{(Adjacent\text{ }side)}^{2}} $
$ \Rightarrow {{(BC)}^{2}}={{(AC)}^{2}}+{{(AB)}^{2}} $
Here, we have,
$\tan \theta =\dfrac{AC}{AB}$
AC = 3
AB = 4
Now, we can write:
$ {{(BC)}^{2}}={{3}^{2}}+{{4}^{2}} $
$\Rightarrow {{(BC)}^{2}}=9+16 $
$ \Rightarrow {{(BC)}^{2}}=25 $
Next, by taking square root on both the sides we get,
$ BC=\sqrt{25} $
$ \Rightarrow BC=5 $
We know that,
$ \sin \theta =\dfrac{Opposite\text{ }side}{Hypotenuse} $
$ \Rightarrow \sin \theta =\dfrac{AC}{BC} $
$ \Rightarrow \sin \theta =\dfrac{3}{5} $
Similarly, we have,
$ \cos \theta =\dfrac{\text{Adjacent }side}{Hypotenuse}$
$ \Rightarrow \cos \theta =\dfrac{AB}{BC} $
$ \Rightarrow \cos \theta =\dfrac{4}{5} $
Now, we have to find ${{\cos }^{2}}\theta -{{\sin }^{2}}\theta $.
By substituting the values of $\sin \theta $ and $\cos \theta $ in the above expression we
get,
$\begin{align}
& \Rightarrow {{\cos }^{2}}\theta -{{\sin }^{2}}\theta ={{\left( \dfrac{4}{5}
\right)}^{2}}-{{\left( \dfrac{3}{5} \right)}^{2}} \\
& \Rightarrow {{\cos }^{2}}\theta -{{\sin }^{2}}\theta =\dfrac{16}{25}-\dfrac{9}{25} \\
\end{align}$
Next, by taking LCM we obtain:
$\begin{align}
& \Rightarrow {{\cos }^{2}}\theta -{{\sin }^{2}}\theta =\dfrac{16-9}{25} \\
& \Rightarrow {{\cos }^{2}}\theta -{{\sin }^{2}}\theta =\dfrac{7}{25} \\
\end{align}$
Therefore, we can say that when $\tan \theta =\dfrac{3}{4}$ ,${{\cos }^{2}}\theta -{{\sin
}^{2}}\theta =\dfrac{7}{25}$.
Hence, the correct answer for this question is option (a).
Note: Here, you must have an idea about the trigonometric ratios and the definitions. Alternatively, the expression ${{\cos }^{2}}\theta -{{\sin }^{2}}\theta $ can also be written as $2{{\cos }^{2}}\theta -1$ or $1-2{{\sin }^{2}}\theta $. So, here you can find any one of the values either $\sin \theta $ or $\cos \theta $ to obtain the answer.
