Question: Prove that: $\dfrac{1 + cos2A}{2} = cos^2A$

Solution:

First Method
Using identity of 1 and formula cos2A = cos²A - sin²A
Taking LHS

$= \dfrac{1 + cos2A}{2}$

$= \dfrac{(cos^2 A + sin^2A) + (cos^2A - sin^2A}{2}$

$= \dfrac{2cos^2A}{2}$

$= cos^2A$

= RHS


Second Method:
Using formula cos2A = 2cos²A -1
Taking LHS

$= \dfrac{1 + cos2A}{2}$

$= \dfrac{1 + (2cos^2A -1)}{2}$

$= \dfrac{2cos^2A}{2}$

$= cos^2A$

= RHS


Third Method:
Using formula 1+cos2A = 2cos²A
Taking LHS

$= \dfrac{1 + cos2A}{2}$

$= \dfrac{2cos^2A}{2}$

$= cos^2A$

= RHS


There are numerous other ways to solve this question as well. If you know, you can put them in the comments box as well.

#Trigonometry
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