Prove the following identity:

Prove the following identity:

Prove the following identity:

What is the value of (Cos²67^o - sin² 23^o)?

If $4 \tan \mathrm{\theta } = 3, \mathrm{evaluate} \left(\frac{4 \sin \mathrm{\theta } - \cos \mathrm{\theta } + 1}{4 \sin \mathrm{\theta } + \cos \mathrm{\theta } - 1}\right)$

If tan 2A = cot (A - 18^o), where 2A is an angle, find the value of A

Prove that $\frac{\sin \mathrm{A} - 2{\sin }^3 \mathrm{A}}{2 {\cos }^3 \mathrm{A} - \cos \mathrm{A}} = \tan \mathrm{A}$.