So I understand that $\sin^2\theta \equiv 1 - \cos^2 \theta$. I am doing homework problems for trigonometric integrals and sometimes they use this formula to put into the equation and sometimes they put $\sin^2{t}=\frac{1}{2}(1-\cos{2t})$.
For what situations do I use each one? I believe there is more identities for $\text{sin}$ to go along with these as well. Let me know please.
Generally, we use the pythagorean identity when having $\cos$ is better than having $\sin$ and there are more trig functions floating around. This may be beneficial when
You want to make a u-substitution e.g. $\int\sin^3x\ dx=\int\sin x(1-\cos^2)\ dx$
Can cancel some terms away using $\sin^2+\cos^2=1$.
Likewise, if $\sin$ is by itself in an integral, you'll want to use the double angle identity. This usually helps to
You will notice that for integrals of the form $\int\sin^nx\ dx$, you'll want to apply the pythagorean identity if $n$ is odd to create a u-substitution scenario and if $n$ is even, you'll want to use the power reducing double angle identity repeatedly until all exponents are odd. Then apply the pythagorean.
My answer will stress the importance of the second of these formulas dealing with angle doubling or halving. I wish that you have already seen matrices, a fundamental tool for dealing with geometry in general and trigonometry in particular.
It is true to say that formulas
$$\tag{1}\cos^2{t}=\frac{1}{2}(1+\cos{2t}) \ \ \ \ \ \ \ \ \text{and} \ \ \ \ \ \ \ \ \sin^2{t}=\frac{1}{2}(1-\cos{2t})$$
and their sister formulas:
$$\tag{2}\cos(2t)=\begin{cases}2\cos^2(t)-1\\1-2\sin^2(t)\\ \cos^2(t)-\sin^2(t)\end{cases} \ \ \ \ \ \ \ \ \text{and} \ \ \ \ \ \ \ \ \sin(2t)=2 \sin(t) \cos(t)$$
are, among, say, a dozen or two dozen trigonometric formulas, the most used, just because they involve doubling/halving of angles, an operation that is very often encountered.
in Physics, for example in Optics.
in Mathematics, let us take 2 examples in plane geometry (among many):
A) the matrix of the (orthogonal) symmetry with respect to line $y=\tan(t)x$ is:
$$\tag{*}S_t=\pmatrix{\cos(2t)& \ \ \sin(2t)\\\sin(2t)&-\cos(2t)}$$
(one can verify that $S_t$ an orthogonal matrix with $\det(S_t)=-1.$)
What is the interest of formula (*) ?
For example showing that the composition of two symmetries: $S_t$ followed by $S_{t'}$ is a rotation $R_{\alpha}$ with angle $\alpha = 2(t'-t)$, the double of the angle between the two axes of symmetry; almost instantaneous proof:
$$\tag{**}S_{t'}S_{t}=\pmatrix{\cos(2t')& \ \ \sin(2t')\\\sin(2t')&-\cos(2t')}\pmatrix{\cos(2t)& \ \ \sin(2t)\\\sin(2t)&-\cos(2t)}=\pmatrix{\cos(\alpha)& \ \ -\sin(\alpha)\\ \sin(\alpha)&\ \ \ \ \ \cos(\alpha)}$$
B) the reduction of conics.
Let us look for the change of variables that will transform the quadratic form
$$\tag{3}Ax^2+2Bxy+Cy^2 \ \ \ \ \ \ \ \ \text{into} \ \ \ \ \ \ \ \ A'X^2\pm B'Y^2$$
(having no longer a "rectangle term").
Let us take the following change of variables, which amounts to a rotation $R_t$:
$$\tag{4}\binom{x}{y}=\pmatrix{\cos(t)& \ \ -\sin(t)\\\sin(t)&\ \ \ \ \ \cos(t)}\binom{X}{Y} \ \ \ \iff \ \ \ \begin{cases}x=X \cos(t) - Y \sin(t)\\y=X \sin(t) + Y \cos(t) \end{cases}.$$
(with unknown $t$). Plugging (4) into (3) gives
$$A(X \cos(t) - Y \sin(t))^2+2B(X \cos(t) - Y \sin(t))(X \sin(t) + Y \cos(t))+C(X \sin(t) + Y \cos(t))^2 $$
$$=\left(...\right) \,X^2 + \left( 2B(\cos^2(t)-\sin^2(t))-(A-C)2cos(t)\sin(t) \right) \,X\,Y + \left( ... \right) \,Y^2$$
(where the dotted coefficients are unimportant at this step).
Suppressing "rectangle" term $XY$ amounts to find $t$ such that
$$ 2B(\cos^2(t)-\sin^2(t))-(A-C)2cos(t)\sin(t)=0$$
Expression that can be transformed into:
$$ 2B\cos(2t)-(A-C)\sin(2t)=0 \ \ \ \iff \ \ \ \tan(2t)=\dfrac{2B}{A-C}$$
$$\iff \ \ \ t=\frac12 atan\left(\dfrac{2B}{A-C}\right)\pm \frac{\pi}{2}$$
It has been possible to find this "unique" value of $t$ thanks to formulas (2)!