The Proof of $\cos\phi=\gamma$ Equation in Special Relativity In the Introductory Special Relativity book, by W. G. V. Rosser, page 182, Section 7.3, the author is defining the 4-vector methods using complex numbers. In his derivation, he writes the following equation (Equation 7.15):
$$\cos\phi=\gamma$$

Equations (7.9) and (7.12) can be rewritten in the form
  $$ X'_1 = X_1 \cos\phi + X_4 \sin\phi, \tag{7.13} $$
$$ X'_4 = -X_1 \sin\phi + X_4 \cos\phi, \tag{7.14} $$
  where
  $$ \cos\phi = \gamma, \tag{7.15} $$
$$ \begin{align}
 \sin\phi &= (1-\cos^2\phi)^{1/2}
 = \left(1-\frac{1}{1-v^2/c^2}\right)^{1/2} \\
 &= \left(\frac{-v^2/c^2}{1-v^2/c^2}\right)^{1/2}
 = \frac{i\gamma v}{c}.
 \end{align} \tag{7.16} $$

The author deduced this equation, but I am not convinced. I would like to know please from where this equation came from? 
 A: It looks like the author is using a convention where the rapidity, $\phi$, is imaginary. The reason for doing this is to ease your introduction to hyperbolic trigonometry. The reason for doing this is because it makes boosts look exactly like rotation.
In 2-dimensions, a rotation matrix has the form
$$\left[\begin{array}{c}
x' \\
y' \end{array}\right] = \left[\begin{array}{cc}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta \end{array}\right]\left[\begin{array}{c}
x \\
y \end{array}\right].$$
In special relativity, a 2-d boost looks like
$$\left[\begin{array}{c}
ct' \\
x' \end{array}\right] = \left[\begin{array}{cc}
\gamma & -\gamma \frac{v}{c} \\
-\gamma \frac{v}{c} & \gamma \end{array}\right]\left[\begin{array}{c}
ct \\
x \end{array}\right], \tag1$$
with $\gamma = \frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^2}}$.
The usual definition given for rapidity $\phi$ that I've seen is to say that $\frac{v}{c}=\tanh\phi$. You can then use some hyperbolic trig identities to get that $\gamma = \cosh\phi$ and $\frac{\gamma v}{c} = \sinh\phi$. Dropping those into (1) you get the classic form of a hyperbolic rotation matrix
$$\left[\begin{array}{c}
ct' \\
x' \end{array}\right] = \left[\begin{array}{cc}
\cosh\phi & -\sinh\phi  \\
-\sinh\phi & \cosh\phi \end{array}\right]\left[\begin{array}{c}
ct \\
x \end{array}\right].\tag2$$
The determinant of the matrix in (2) is 1, as required for a "rotation", it just happens to preserve $x^2 - (ct)^2$ instead of Euclidean length.
You can make (2) look like an ordinary rotation matrix if you use the substitution $\phi\rightarrow i\phi$ and $ct \rightarrow -i x^4$ (may need to tweak the signs).
And that's what the book in question did.
