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I'm currently studying shockwaves, particularly their effects on drag. I've stumbled upon an odd formula in "Elements of Gasdynamics" (Liepman and Roshko, pages 52-53), in there I saw an alternative formula for the theta-beta-mach diagram, and the author committed a passage in which I've been having problems replicating.

Starting from: $ \frac{\tan (\beta - \theta )}{\tan(\beta)} = \frac {(\gamma-1) M_1^2 \sin^2(\beta)+2}{(\gamma+1)M_1^2\sin^2(\beta)} $

He divides and multiplies right hand side by $\frac{1}{2} M_1^2 \sin^2(\beta)$

To obtain: $\frac{1}{M_1^2 \sin^2(\beta) } = \frac{\gamma+1}{2}\frac{tan(\beta-\theta)}{\tan{\beta}}-\frac{\gamma-1}{2}$

Finally he ends up with: $M_1^2 \sin^2(\beta) -1 = \frac{\gamma+1}{2} M_1^2 \frac{\sin(\beta)\sin(\theta)}{\cos(\beta-\theta)}$

The last passage is not very clear and I have no idea how he ended up with that, I've tried with Wolfram and both equations are solved for the same* points, yet I am unable to find a way to obtain the latter. My main question is if there is other literature where this formula appears or a way to solve it, I liked it because it allows the introduction to linearized supersonic pretty easily rather than using the potential. Needless to say I tried all the trigonometric tricks I could find. Any help is appreciated, thanks.

PS: I consulted, Fundaments of AD, Elements of Aerodynamics of Supersonic Flows, aerodynamics for engineering students and Dyamics and Thermodynamics of Compressible Flows.

EDIT: I found it in NACA Report 1135

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  • $\begingroup$ I'll give you a hint: $1 + \tan{x} \tan{y} = \tfrac{ \cos{ \left( x - y \right) } }{ \cos{x} \cos{y} }$ $\endgroup$
    – honeste_vivere
    Commented Oct 24, 2023 at 13:50
  • $\begingroup$ I pulled out my original edition of Liepmann and Roshko (c. 1957, 1965 printing) and found the equations above on pp. 87-88. Getting the second equation is easy enough, but I am stumped also on getting the 3rd. The 2nd equation is also in Fundamentals of Gas Dynamics by Jerzy A. Owczarek, copyright 1964 on page 177 (with different formatting). $\endgroup$
    – W H G
    Commented Oct 25, 2023 at 1:16

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