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Mathematics Question on Vector Algebra

Show that
(a⃗−b⃗)×(a⃗+b⃗)=2(a⃗×b⃗)(\vec{a}-\vec{b})\times(\vec{a}+\vec{b})=2(\vec{a}\times \vec{b})

Answer

(a⃗−b⃗)×(a⃗+b⃗)(\vec{a}-\vec{b})\times(\vec{a}+\vec{b})
=(a⃗−b⃗)×a⃗+(a⃗−b⃗)×b⃗=(\vec{a}-\vec{b})\times \vec{a}+(\vec{a}-\vec{b})\times \vec{b} [By distributivity of vector product over addition]
=a⃗×a⃗−b⃗×a⃗+a⃗×b⃗−b⃗×b⃗=\vec{a}\times \vec{a}-\vec{b}\times\vec{a}+\vec{a}\times\vec{b}-\vec{b}\times\vec{b} [Again,by distributivity of vector product over addition]
=0⃗+a⃗×b⃗+a⃗×b⃗−0⃗=\vec{0}+\vec{a}\times\vec{b}+\vec{a}\times\vec{b}-\vec{0}
=2a⃗×b⃗=2\vec{a}\times\vec{b}