Here are a few concepts/ identities which we feel are important for exams:-
♦ Value of equation of type:- \( cosθ.cos2θ.cos4θ.cos8θ ....cos2^{n-1}θ \) is \({sin2^{n}θ\over 2^n.Sinθ.}\) ; where n is the number of terms
If you do not want to learn this formula, just understand the concept behind it, as many times direct questions are asked from this pattern.
Concept: multiply & divide the equation of this format by 2sinθ
⇒ \({2sinθ .cosθ.cos2θ.cos4θ.cos8θ ....cos2^{n-1}θ }\over 2sinθ \)
we already know that sin2θ = 2.sinθ.cosθ, replacing in the above equation, we get
⇒ \({sin2θ .cos2θ.cos4θ.cos8θ ....cos2^{n-1}θ }\over 2sinθ \)
this time we: multiply & divide the equation by 2sin2θ & thus the value of angle doubles each time and forms a chain and at the end we get
⇒ \({sin2^{n}θ\over 2^n.Sinθ.}\)
♦ Value of equation type is tanθ.tan(60° + θ).tan(60° - θ) = tan3θ, you can use tan(A + B) identity to prove it, however, the derivation is not important, you can also check validity by assuming θ° = 15°