Conditions on $a,b,c,d$ such that $\gamma (t)$ is regular for all $t$? 
I solved an exercise in my book and I was wondering if someone could
  look at my answer and tell me if it is correct please?

The exercise is this: Let $\gamma (t) = (a \cos t + b \sin t, c \cos t + d \sin t)$. What are the conditions on $a,b,c,d$ so that $\gamma$ is regular? What is the equation of the curve when it is regular? What happens for other values of $a,b,c,d$?
My answer:
The derivative is given by $\gamma' (t) = (b \cos t - a \sin t , d \cos t - c \sin t)$ and the square of its norm is $(b \cos t - a \sin t)^2 + (d \cos t - c \sin t)^2$. 
Therefore the norm is zero iff both terms are zero. The first term is zero iff
$$ b \cos t = a \sin t$$
But any scaled cosine curve has to cross any scaled sine curve at some point in time so to avoid that this equality never occurs either $b$ or $a$ has to be zero (while the other one has to be non-zero). Similarly for the second term. 
As a consequence, the curve has to either be an ellipse or a line. 
 A: Hints: Keeping track of contingencies "naively" looks vexing. Here are a couple of approaches, one consisting of a standard, general-purpose trig trick, the other an algebraic observation for this particular curve.

If $(a, b) = (0, 0)$ or $(c, d) = (0, 0)$, the curve is obviously not regular. Otherwise, there exist real numbers $\phi_{1}$ and $\phi_{2}$, unique up to added integer multiples of $2\pi$, such that
$$
\frac{(a, -b)}{\sqrt{a^{2}+ b^{2}}} = (\cos\phi_{1}, \sin\phi_{1}),\qquad
\frac{(c, -d)}{\sqrt{c^{2}+ d^{2}}} = (\cos\phi_{2}, \sin\phi_{2}),
$$
and the sum formulas for the circular functions give
$$
\gamma(t) = \left(\sqrt{a^{2} + b^{2}} \cos(t + \phi_{1}), \sqrt{c^{2} + d^{2}} \cos(t + \phi_{2})\right).
$$
Since the radicands are positive, $\gamma$ is regular if and only if
$$
\tilde{\gamma}(t) = \left(\cos t, \cos(t - \phi_{1} + \phi_{2})\right)
$$
is regular. It's now straightforward to determine when the velocity is zero.

Alternatively, for this particular curve you might notice that
$$
\gamma(t) = (\cos t)(a, c) + (\sin t)(b, d)
  = \left[\begin{array}{@{}cc@{}}
  a & b \\
  c & d \\
  \end{array}\right]\left[\begin{array}{@{}c@{}}
    \cos t \\
    \sin t
  \end{array}\right]
$$
is a "trigonometric linear combination" of a pair of vectors, a.k.a., the image of the unit circle (a regular curve) under the linear transformation with indicated matrix. Consequently, $\gamma$ is regular if and only if the matrix has rank $2$. (Why?)
This is probably the approach intended by the book's author, since it's easy to describe the image of $\gamma$ geometrically from knowledge of linear transformations.
