If | cosecα 1 0 1 2 cosecα 1 0 1 2 cosecα |=(1)/(2)(z^3+(1

step 1 I have we given here that the determinants here equal to 1 over 2 z cube plus 1 over z cube. We'

If | cosecα 1 0 1 2 cosecα 1 0...

If $\left|\begin{array}{ccc}\operatorname{cosec} \alpha & 1 & 0 \\ 1 & 2 \operatorname{cosec} \alpha & 1 \\ 0 & 1 & 2 \operatorname{cosec} \alpha\end{array}\right|=\frac{1}{2}\left(z^{3}+\frac{1}{z^{3}}\right)$, then $z$ is equal to (A) $\sin \alpha / 2$ (B) $\cos \alpha / 2$ (C) $\tan \alpha / 2$ (D) None of these

If $\left|\begin{array}{ccc}\operatorname{cosec} \alpha & 1 & 0 \\ 1 & 2 \operatorname{cosec} \alpha & 1 \\ 0 & 1 & 2 \operatorname{cosec} \alpha\end{array}\right|=\frac{1}{2}\left(z^{3}+\frac{1}{z^{3}}\right)$,
then $z$ is equal to
(A) $\sin \alpha / 2$
(B) $\cos \alpha / 2$
(C) $\tan \alpha / 2$
(D) None of these

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Key Concepts

Trigonometric Functions and Identities

Trigonometric functions such as sine, cosine, and their reciprocals are essential in relating angles to ratios in geometric contexts. Their identities, including sum-to-product, double-angle, and half-angle formulas, play a key role in transforming and simplifying expressions, especially when algebraic and trigonometric expressions are combined.

Half-Angle Formulas

Half-angle formulas allow for expressing trigonometric functions of half of an angle in terms of functions of the full angle. These formulas are particularly useful for simplifying complex trigonometric expressions and solving equations involving trigonometric functions, serving as an important bridge in linking different forms of trigonometric relationships.

Matrix Determinants

Matrix determinants are scalar quantities computed from square matrices that provide insights into the properties of linear transformations, such as invertibility, volume scaling, and system solvability. They can be calculated using methods like cofactor expansion or reduction to triangular form and are fundamental in linear algebra.

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