Question

If θ = 30°, verify that

(i) $\tan 2\mathrm{\theta }=\frac{2\tan \mathrm{\theta }}{1-{\tan }^2\mathrm{\theta }}$
(ii) $\sin 2\mathrm{\theta }=\frac{2\tan \mathrm{\theta }}{1+{\tan }^2\mathrm{\theta }}$
(iii) $\cos 2\mathrm{\theta }=\frac{1-{\tan }^2\mathrm{\theta }}{1+{\tan }^2\mathrm{\theta }}$
(iv) cos 3θ = 4 cos³ θ − 3 cos θ

Answer 1

(i) Given:

…… (1)

To verify:

…… (2)

Now consider LHS of the expression to be verified in equation (2)

Therefore,

Now by substituting the value of from equation (1) in the above expression

We get,

Now by substituting the value of from equation (1) in the expression

We get,

…… (4)

Now by comparing equation (3) and (4)

We get,

Hence

(ii) Given:

…… (1)

To verify:

…… (2)

Now consider right hand side

Hence it is verified that,

(iii) Given:

…… (1)

To verify:

…… (2)

Now consider left hand side of the equation (2)

Therefore,

Now consider right hand side of equation (2)

Therefore,

Hence it is verified that,

(iv) Given:

…… (1)

To verify:

…… (2)

Now consider left hand side of the expression in equation (2)

Therefore

Now consider right hand side of the expression to be verified in equation (2)

Therefore,

Hence it is verified that,