How can we prove it using vectors?

Given \[a^{2} + b^{2} + 2ab \cos \theta = 1 , c^{2} + d^{2} + 2cd \cos \theta = 1\] and \[(ac + bd) + ( ad + bc) \cos \theta = 0\]

Then prove that $a^{2} + c^{2} = cosec^{2} \theta$

Im struggling to prove this for a while.... Can anyone please post their proofs in the comments section? Thanks!

#Algebra

Easy Math Editor

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$