Evaluate the following trigonometric equation $\dfrac{\tan 65{}^\circ }{\cot 25{}^\circ }$ .
Answer
326.1k+ views
Hint: Use conversion of trigonometric functions of cot and tan by changing their angles. Don’t go for calculating exact values of $\tan 65{}^\circ $ and $\cot 25{}^\circ $. So, convert and simplify.
Complete step-by-step answer:
Here, we need to evaluate the value of the expression $\dfrac{\tan 65{}^\circ }{\cot 25{}^\circ }$ .
As we do not know the exact value of $\tan 65{}^\circ $ and $\cot 25{}^\circ $ , so we cannot solve the expression by putting the values of $\tan 65{}^\circ $ or $\cot 25{}^\circ $. And if we go for finding values of $\tan 65{}^\circ $ and $\cot 25{}^\circ $that would be very complex and it is highly possible that we will not get exact values of them.
Hence, we need to use some relationship in $\tan \theta $ and $\cot \theta $ i.e. we need to convert $\tan \theta $ to $\cot \theta $ or vice-versa by using some identity.
As we have already learnt that we can convert one trigonometric function to another by adding $90{}^\circ $ or $180{}^\circ $ to the angle involved in the function or subtracting as well.
So, we know the complementary conversions of $\tan \theta $ to $\cot \theta $, $\sin \theta $ to $\cos \theta $, $\csc \theta $ to $\sec \theta $ or vice-versa by using identities as
$\tan \left( 90-\theta \right)=\cot \theta $ or $\cot \left( 90-\theta \right)=\tan \theta $
$\cos \left( 90-\theta \right)=\sin \theta $ or $\sin \left( 90-\theta \right)=\cos \theta $
$\sec \left( 90-\theta \right)=\csc \theta $ or $\csc \left( 90-\theta \right)=\sec \theta $
As, we have only tan and cot functions and summation of given angles i.e. 65 and 25 is $90{}^\circ $ . So, we can convert $\tan 65{}^\circ $ to cot function by the following approach.
As, we can write $\tan 65{}^\circ $ as $\tan \left( 90{}^\circ -25{}^\circ \right)$ .
Now, we can write $\tan \left( 90{}^\circ -25{}^\circ \right)$ as $\cot 25{}^\circ $ from the above identity $\tan \left( 90-\theta \right)=\cot \theta $ .
Hence, we get the given expression as \[\dfrac{\tan 65{}^\circ }{\cot 25{}^\circ }=\dfrac{\cot 25{}^\circ }{\cot 25{}^\circ }=1 .\]
Hence, the answer is ‘1’.
Note: One can waste a lot of time with the trigonometric identities if he/she may go for finding exact values of $\tan 65{}^\circ $ and $\cot 25{}^\circ $. Observation of sum of both angles $65~{}^\circ +25{}^\circ =90{}^\circ $ is the key point of the question. One cannot convert $65{}^\circ $ to $180{}^\circ -115{}^\circ $ or $25{}^\circ $ to $180{}^\circ -155{}^\circ $ and now try to apply identities of $\tan \left( 180-\theta \right)$ or $\cot \left( 180-\theta \right)$ which will not give the answer. Hence writing with subtraction of $90{}^\circ $ to any of the functions is the only way to get the exact answer.
Complete step-by-step answer:
Here, we need to evaluate the value of the expression $\dfrac{\tan 65{}^\circ }{\cot 25{}^\circ }$ .
As we do not know the exact value of $\tan 65{}^\circ $ and $\cot 25{}^\circ $ , so we cannot solve the expression by putting the values of $\tan 65{}^\circ $ or $\cot 25{}^\circ $. And if we go for finding values of $\tan 65{}^\circ $ and $\cot 25{}^\circ $that would be very complex and it is highly possible that we will not get exact values of them.
Hence, we need to use some relationship in $\tan \theta $ and $\cot \theta $ i.e. we need to convert $\tan \theta $ to $\cot \theta $ or vice-versa by using some identity.
As we have already learnt that we can convert one trigonometric function to another by adding $90{}^\circ $ or $180{}^\circ $ to the angle involved in the function or subtracting as well.
So, we know the complementary conversions of $\tan \theta $ to $\cot \theta $, $\sin \theta $ to $\cos \theta $, $\csc \theta $ to $\sec \theta $ or vice-versa by using identities as
$\tan \left( 90-\theta \right)=\cot \theta $ or $\cot \left( 90-\theta \right)=\tan \theta $
$\cos \left( 90-\theta \right)=\sin \theta $ or $\sin \left( 90-\theta \right)=\cos \theta $
$\sec \left( 90-\theta \right)=\csc \theta $ or $\csc \left( 90-\theta \right)=\sec \theta $
As, we have only tan and cot functions and summation of given angles i.e. 65 and 25 is $90{}^\circ $ . So, we can convert $\tan 65{}^\circ $ to cot function by the following approach.
As, we can write $\tan 65{}^\circ $ as $\tan \left( 90{}^\circ -25{}^\circ \right)$ .
Now, we can write $\tan \left( 90{}^\circ -25{}^\circ \right)$ as $\cot 25{}^\circ $ from the above identity $\tan \left( 90-\theta \right)=\cot \theta $ .
Hence, we get the given expression as \[\dfrac{\tan 65{}^\circ }{\cot 25{}^\circ }=\dfrac{\cot 25{}^\circ }{\cot 25{}^\circ }=1 .\]
Hence, the answer is ‘1’.
Note: One can waste a lot of time with the trigonometric identities if he/she may go for finding exact values of $\tan 65{}^\circ $ and $\cot 25{}^\circ $. Observation of sum of both angles $65~{}^\circ +25{}^\circ =90{}^\circ $ is the key point of the question. One cannot convert $65{}^\circ $ to $180{}^\circ -115{}^\circ $ or $25{}^\circ $ to $180{}^\circ -155{}^\circ $ and now try to apply identities of $\tan \left( 180-\theta \right)$ or $\cot \left( 180-\theta \right)$ which will not give the answer. Hence writing with subtraction of $90{}^\circ $ to any of the functions is the only way to get the exact answer.
Last updated date: 03rd Jun 2023
•
Total views: 326.1k
•
Views today: 7.82k
Recently Updated Pages
If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE
