Courses
Courses for Kids
Free study material
Offline Centres
More
Store Icon
Store

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)$\dfrac{4}{25}$

seo-qna
Last updated date: 27th Mar 2024
Total views: 404.4k
Views today: 10.04k
MVSAT 2024
Answer
VerifiedVerified
404.4k+ views

Hint: 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,

seo images


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.



Recently Updated Pages