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If x = ( - ) . ( - ), y = ( - ) . ( - ) and z

Question

If $x = sin (α - β) . sin (γ - δ), y = sin (β - γ) . sin (α - δ)$ and
$z = sin (γ - α) . sin (β - δ),$ then

x + y + z = 0

x + y - z = 0

y + z - x = 0

x^{3} + y^{3} + z^{3} = 3 x yz

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Jack Discussed

x = sin (α - β) . sin (γ - δ), y = sin (β - γ) . sin (α - δ)

and

z = sin (γ - α) . sin (β - δ),

then

8 mins ago

Discuss this question LIVE

8 mins ago

Text solutionVerified

x = sin (α - β) sin (γ - δ) = (sin α cos β - cos α sin β) (sin γ cos δ - cos γ sin δ)

= sin α cos β sin γ cos δ - sin α cos β cos γ sin δ - cos α sin β sin γ cos δ + cos α sin β cos γ sin δ

y = sin (β - γ) sin (α - δ) = (sin β cos γ - cos α sin γ) (sin α cos - cos α sin)

= sin β cos γ sin α cos δ - sin β cos γ cos α sin δ - cos β sin γ sin α cos δ + cos β sin γ cos α sin δ

z = sin (γ - α) sin (β - δ) = (sin γ cos - cos γ sin) (sin cos - cos sin)

= sin γ cos α sin β cos δ - sin γ cos α cos β sin δ - cos γ sin α sin β cos δ + cos γ sin α cos β sin δ

\Rightarrow x + y + z = 0

Using

a + b + c = 0 \Rightarrow a^{3} + b^{3} + c^{3} = 3 ab c

We get

x^{3} + y^{3} + z^{3} = 3 x yz

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Question Text	If $x = sin (α - β) . sin (γ - δ), y = sin (β - γ) . sin (α - δ)$ and $z = sin (γ - α) . sin (β - δ),$ then
Answer Type	Text solution:1
Upvotes	150