Question

If $\sec A=\frac{5}{4}$, verify that $\frac{3\sin A-4{\sin }^{3}A}{4{\cos }^{3}A-3\cos A}=\frac{3\tan A-{\tan }^{3}A}{1-3{\tan }^{2}A}$.

Answer 1

Given:

…… (1)

To verify:

…… (2)

Now we know that

Therefore

Now, by substituting the value of from equation (1)

We get,

Therefore,

…… (3)

Now, we know the following trigonometric identity

Therefore,

Now by substituting the value of from equation (3)

We get,

Now by taking L.C.M

We get,

Now, by taking square root on both sides

We get,

Therefore,

…… (4)

Now, we know that

Now by substituting the value of and from equation (3) and (4) respectively

We get,

Therefore

…… (5)

Now from the expression of equation (2)

Now by substituting the value of and from equation (3) and (4)

We get,

Therefore,

Now by taking L.C.M of both numerator and denominator

We get,

…… (6)

Now from the expression of equation (2)

Now by substituting the value of from equation (5)

We get,

Now by taking L.C.M

We get,

Now,

Therefore,

Therefore,

…… (7)

Now by comparing equation (6) and (7)

We get,