Cos^4 (A/2)- sin^4(A/2)=cosA

LHS=cos^4(A/2)-sin^4(A/2)

={cos^2(A/2)}^2 -{sin^2(A/2)}^2

={cos^2(A/2) - sin^2(A/2)}{cos^2(A/2) +sin^2(A/2)}

=cosA×1

=cosA

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.