Find the value of (i) [(sin^2 x)/(2)] (ii) [sin^2((π)/(2)) x] (iii) [cos^2((π)/(2)) x] (iv) [(co

step 1 Hi guys, so we have this problem in which we are given the question in four parts. The first par

Find the value of (i) [(sin^2 x)/(2)] (ii) [sin^2((π)/(2))...

Find the value of
(i) $\left[\frac{\sin ^{2} x}{2}\right]$
(ii) $\left[\sin ^{2}\left(\frac{\pi}{2}\right) x\right]$
(iii) $\left[\cos ^{2}\left(\frac{\pi}{2}\right) x\right]$
(iv) $\left[\frac{\cos ^{2} x}{2014}\right]$.

Key Concepts

Greatest Integer Function (Floor Function)

The greatest integer function assigns to any real number the largest integer that is less than or equal to that number. This function, often denoted by square brackets, plays a key role in problems where outcomes are reduced to their integer component after evaluating possibly continuous or fractional expressions.

Range of Squared Trigonometric Functions

Squaring sine or cosine functions confines their values within the interval [0, 1] regardless of the original angle. This property is critical in assessing expressions involving sin²x or cos²x, particularly when these expressions are further modified by division or the floor function.

Evaluation at Special Trigonometric Angles

Trigonometric functions take on well-known values at specific angles; for instance, sin(?/2) equals 1 and cos(?/2) equals 0. These standard values simplify the evaluation of expressions when trigonometric functions are computed at such angles.

Scaling Effects in Mathematical Expressions

Applying a constant factor, either as a multiplier or divisor, to function outputs alters the value of the expression while preserving certain properties like ordering. This concept is particularly significant when such scaled expressions are combined with the greatest integer function, as the resulting integer value depends on the magnitude of the scaling.

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