simplfy the expression by double angle formula or a half angle formula.
cos^2(x/2)-sin^2(x/2)
2sin(x/2)cos(x/2)
To simplify the expression cos^2(x/2) - sin^2(x/2) using a half-angle formula, we can use the identity:
cos^2(x/2) - sin^2(x/2) = 2sin(x/2)cos(x/2)
Therefore, the simplified expression is 2sin(x/2)cos(x/2).
To simplify the expression cos^2(x/2) - sin^2(x/2) using the double angle formula or a half angle formula, we can use the identity:
cos(2θ) = cos^2(θ) - sin^2(θ).
In this case, we have cos^2(x/2) - sin^2(x/2). Let's use the half angle formula to rewrite it as a double angle:
cos(2(x/4)) = cos^2(x/4) - sin^2(x/4).
Now, let's simplify each term separately using the half angle formulas:
cos^2(x/4) = (1 + cos(x/2)) / 2, since cos(2θ) = 2cos^2(θ) - 1.
sin^2(x/4) = (1 - cos(x/2)) / 2, since sin(2θ) = 2sin(θ)cos(θ).
Substituting these values back into the expression, we get:
(1 + cos(x/2)) / 2 - (1 - cos(x/2)) / 2.
Simplifying this further, we obtain:
(1 + cos(x/2) - 1 + cos(x/2)) / 2.
Combining like terms, we end up with:
2cos(x/2) / 2.
Finally, we cancel out the 2's to get the simplified expression:
cos(x/2).
Overall, cos^2(x/2) - sin^2(x/2) simplifies to cos(x/2) using the half angle formula.