Question

Evaluate :- 2/3 cosec^2 58° - 2/3 cot 58 tan 32 - 5/3 tan 13° tan 37° tan 45° tan 53° tan 77°

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Answer 1

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Answer 2

-1 is the value of $\bold{\frac{2}{3} \csc ^{2} 58^{\circ}-\frac{2}{3} \cot 58^{\circ} \tan 32^{\circ}-\frac{5}{3} \tan 13^{\circ} \tan 37^{\circ} \tan 45^{\circ} \tan 53^{\circ} \tan 77^{\circ}}$

Given:

The given expression is

$\frac{2}{3} \csc ^{2} 58^{\circ}-\frac{2}{3} \cot 58^{\circ} \tan 32^{\circ}-\frac{5}{3} \tan 13^{\circ} \tan 37^{\circ} \tan 45^{\circ} \tan 53^{\circ} \tan 77^{\circ}$

To find:

The value of  $\frac{2}{3} \csc ^{2} 58^{\circ}-\frac{2}{3} \cot 58^{\circ} \tan 32^{\circ}-\frac{5}{3} \tan 13^{\circ} \tan 37^{\circ} \tan 45^{\circ} \tan 53^{\circ} \tan 77^{\circ} = ?$

Solution:

We know that the sum of complementary angles is $90^\circ$.

So,$13+77=90 ; 45+45 =90 ; 53+37=90$

Let us consider $tan13 tan37 tan45 tan53 tan77$ from the given expression

$\begin{array}{l}{=\tan 13 \tan 37 \tan 45 \tan 53 \tan 77} \\ {=\tan 13 \tan 37(1) \tan 53 \tan 77 \quad\left(\text { since tan45 }=\frac{\sin 45}{\cos 45}=\frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}}\right)} \\ {=1}\end{array}$

Now, let us consider the given expression

$\begin{array}{l}{\frac{2}{3} \csc ^{2} 58^{\circ}-\frac{2}{3} \cot 58^{\circ} \tan 32^{\circ}-\frac{5}{3} \tan 13^{\circ} \tan 47^{\circ} \tan 45^{\circ} \tan 53^{\circ} \tan 77^{\circ}} \\ {=\frac{2}{3} \csc ^{2} 58^{\circ}-\frac{2}{3} \cot 58^{\circ} \tan 32-\frac{5}{3} \times 1} \\ {=\frac{2}{3} \csc ^{2} 58-\frac{2}{3} \cot 58 \tan (90-58)-\frac{5}{3}}\end{array}$

$\begin{array}{l}{=\frac{2}{3} \csc ^{2} 58-\frac{2}{3} \cot 58 \cot 58-\frac{5}{3}} \\ {=\frac{2}{3}\left(\csc ^{2} 58-\cot ^{2} 58\right)-\frac{5}{3} |} \\ {=\frac{2}{3}(1)-\frac{5}{3} \quad\left(A s \csc ^{2} x-\cot ^{2} x=1\right)} \\ {=\frac{2}{3}-\frac{5}{3}} \\ {=-\frac{3}{3}}\end{array}$

$= -1$

Therefore, $\bold{\frac{2}{3} \csc ^{2} 58^{\circ}-\frac{2}{3} \cot 58^{\circ} \tan 32^{\circ}-\frac{5}{3} \tan 13^{\circ} \tan 37^{\circ} \tan 45^{\circ} \tan 53^{\circ} \tan 77^{\circ} = -1}$