Final answer:
To prove the equation 2sin(π/4-a) * sin(3π/4-a) = cos2a, we apply trigonometric identities and properties. Simplification shows that the equation is true.
Step-by-step explanation:
To prove that 2sin(π/4-a) * sin(3π/4-a) = cos2a, we can use some trigonometric identities and properties. Let's start with the right side of the equation, cos2a.
Using the identity cos2a = cos^2a - sin^2a, we have cos2a = (cos^2a)(1 - sin^2a).
Next, we can substitute sin^2a = 1 - cos^2a from another trigonometric identity. This gives us cos2a = (cos^2a)(1 - (1 - cos^2a)).
Simplifying this expression, we get cos2a = cos^2a - (cos^2a)(1 - cos^2a).
Using the difference of squares, we have cos2a = (cos^2a)(1 - cos^2a) - (cos^2a)(cos^2a).
Combining like terms, we get cos2a = (cos^2a)(1 - cos^2a - cos^2a).
Finally, simplifying further, we obtain cos2a = (cos^2a)(1 - 2cos^2a) which matches the left side of the equation.