Question

If $\cos \theta =\frac{5}{13}$, find the value of $\frac{{\sin }^2\theta -{\cos }^2\theta }{2\sin \theta \cos \theta }\times \frac{1}{{\tan }^2\theta }$.

Answer 1

Given: ……(1)

To Find:

The value of expression

Now, we know that

…… (2)

Now when we compare equation (1) and (2)

We get,

= 512

And

Hypotenuse = 13

Therefore, Triangle representing angle is as shown below

Perpendicular side AB is unknown and it can be found by using Pythagoras theorem

Therefore by applying Pythagoras theorem

We get,

Therefore by substituting the values of known sides

We get,

Therefore,

Therefore,

…… (3)

Now, we know that

Now from figure (a)

We get,

Therefore from figure (a) and equation (3) ,

…… (4)

Now we know that,

Therefore, substituting the value of and from equation (1) and (4)

We get,

Therefore 13 gets cancelled and we get

…… (5)

Now we substitute the value of, and from equation (1) , (4) and (5) respectively in the expression below

Therefore,

We get,

Therefore by further simplifying we get,

Now 169 gets cancelled and gets reduced to

Therefore

Therefore the value of is

That is