Final answer:
While half-angle identities provide a means to express sine and cosine functions of half angles, csc²(x/2) cannot be directly simplified to sec²(x/2) via these identities without additional context or values for x.
Step-by-step explanation:
The correct answer is b) sec²(x/2).To simplify csc²(x/π) using a half-angle identity, we can use the identity:csc²θ = 1 + cot²θUsing this identity with x/π as θ, we have:csc²(x/π) = 1 + cot²(x/π)Since cot²θ is equivalent to cosec²θ - 1, we can substitute cot²(x/π) with csc²(x/π) - 1:csc²(x/π) = 1 + (csc²(x/π) - 1)Combining like terms, we get:csc²(x/π) = csc²(x/π)Therefore, the simplified form of csc²(x/π) is b) sec²(x/2)The correct option for simplifying csc²(x/2using a half-angle identity is option (b) sec²(x/2).Half-angle identities are useful in trigonometry to simplify expressions of the sine, cosine, or tangent functions of half angles.
These identities can be derived from the double-angle formulas. To simplify csc²(x/2), we look for an identity involving the cosecant function or its reciprocal, the sine function. The half-angle formula for sine is sin(x/2) = ±√((1 - cos(x))/2), and since csc(x) = 1/sin(x), we can use the identity related to cosine to find the expression in terms of sec(x), which is 1/cos(x). However, without the specific trigonometric values for x, we can't directly simplify csc²(x/2) to sec²(x/2). It is necessary to note that simply using half-angle formulas will not convert csc directly to sec, as they are reciprocals of different functions and are based on sine and cosine respectively.