Question

If $\tan \theta =\frac{1}{\sqrt{7}}$, show that $\frac{{\mathrm{cosec}}^2\theta -{\sec }^2\theta }{{\mathrm{cosec}}^2\theta +{\sec }^2\theta }=\frac{3}{4}$

Answer 1

Given: ……(1)

To show that

Now, we know that

Since ……(2)

Therefore,

Comparing Equation (1) and (2)

We get,

Therefore, Triangle representing angle is as shown below

Hypotenuse AC 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, from equation (4)

We get,

Therefore,

…… (5)

Now, we know that

Now from figure (a)

We get,

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

…… (6)

Now, we know that

Therefore, from equation (6)

We get,

Therefore,

…… (7)

Now, L.H.S of the equation to be proved is as follows

Substituting the value of and from equation (6) and (7)

We get,

Now by taking L.C.M. in numerator as well as denominator

We get,

Therefore,

Therefore,

Therefore,

Hence proved that