Question

If $\sin \theta =\frac{3}{4}$, prove that $\sqrt{\frac{{\mathrm{cosec}}^2\theta -{\cot }^2\theta }{{\sec }^2\theta -1}}=\frac{\sqrt{7}}{3}$.

Answer 1

Given:

…… (1)

To prove:

…… (2)

By definition,

…… (3)

By Comparing (1) and (3)

We get,

Perpendicular side = 3 and

Hypotenuse = 4

Side BC is unknown.

So we find BC by applying Pythagoras theorem to right angled,

Hence,

Now we substitute the value of perpendicular side (AB) and hypotenuse (AC) and get the base side (BC)

Therefore,

Hence, Base side BC = …… (3)

Now,

Therefore from fig. a and equation (3)

Therefore,

…… (4)

Now,

Therefore from fig. a and equation (1) ,

…… (5)

Now,

Therefore from fig. a and equation (4) ,

…… (6)

Now,

Therefore by substituting the values from equation (1) and (4) ,

We get,

Therefore,

…… (7)

Now by substituting the value of , and from equation (5) ,(6) and (7) respectively in the L.H.S of expression (2) ,

We get,

Therefore,

Hence it is proved that