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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.

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