MHB How to Prove a Trigonometric Identity Involving x, y, and z?

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To prove the identity (x^2-1)(y^2-1)/xy + (y^2-1)(z^2-1)/yz + (z^2-1)(x^2-1)/zx = 4 under the condition xy + yz + zx = 1, trigonometric identities can be utilized, such as cotAcotB + cotBcotC + cotCcotA = 1. Users are encouraged to share their progress or initial thoughts to facilitate better assistance from others. This approach helps identify where they may be struggling or misapplying concepts. Engaging with the community can lead to more effective solutions and insights. Collaboration is key in tackling complex trigonometric proofs.
skcollins
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If xy+yz+zx=1 then prove (x^2-1)(y^2-1)/xy+(y^2-1)(z^2-1)/yz+(z^2-1)(x^2-1)/zx=4 with trigonometric identities such as cotAcotB+cotBcotC+cotCcotA=1
 
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Hello skcollins and welcome to MHB! :D

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Can you post what you have done so far?
 
Thread 'Erroneously finding discrepancy in transpose rule'

Thread 'Erroneously finding discrepancy in transpose rule'

Obviously, there is something elementary I am missing here. To form the transpose of a matrix, one exchanges rows and columns, so the transpose of a scalar, considered as (or isomorphic to) a one-entry matrix, should stay the same, including if the scalar is a complex number. On the other hand, in the isomorphism between the complex plane and the real plane, a complex number a+bi corresponds to a matrix in the real plane; taking the transpose we get which then corresponds to a-bi...
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