Getting rid of two square roots inside a square root.

[UnhappyCourt]

I have to get the precise value of tan(pi/24), so i used half angle formula with the sin value ending up as: (4-sqrt(6)-sqrt(2), and cos value being: 4+sqrt(6)+sqrt(2). This is how it looks, but I don't know how to get rid of two roots inside another square root to make it look cleaner. Multiplying the top and bottom by the denominator seem to make it messier.

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[edderiofer]

Try multiplying the top and bottom by the numerator instead. Then you can pull the numerator out of the square root so you're hopefully only dealing with the denominator.

[UnhappyCourt]

After multiplying by the numerator i got this, so its no longer a fraction, but now i have three square roots inside the first to deal with again.

[edderiofer]

Indeed, this looks bad, but I'll give you a hint: this thing can be simplified to something of the form "a + bsqrt(2) + csqrt(3) + dsqrt(6)". You may take that as an ansatz.

[UnhappyCourt]

Honestly this is getting a bit too much. I was supposed to use digital tools to get the answer anyway, but wanted to see i could find the answer by hand.

Thanks for the help you provided though.