How to express trigonometric equation in terms of of given trigonometric function?

Question

How can I express a trigonometric equation / identity in terms of a given trigonometric function?

using following trigonometric identities

Sin[x]^2+Cos[x]^2==1
Sin[x]/Cos[x]==Tan[x]
Csc[x]==1/Sin[x]
Sec[x]==1/Cos[x]
Cot[x]==1/Tan[x]

Examples

$$\text{convert}(\sin x,\cos)\Rightarrow \pm\sqrt{1-\cos^2(x)}$$ $$\text{convert}(\cos x,\sin)\Rightarrow \pm\sqrt{1-\sin^2(x)}$$ $$\text{convert}\left(\frac{\cos x}{\sin x},\tan\right)\Rightarrow\frac{1}{\tan x}$$

convert[eqn_,trigFunc_]:=??

Answer 1

This is a new version of my answer in response to the edited question (the first version is here).

It is based on the same idea, but the Weierstrass substitution rules are now generated by Mathematica (instead of entered by hand) and results with $\pm$ solutions are correctly returned.

First, generate the Weierstrass substitution rules

$TrigFns = {Sin, Cos, Tan, Csc, Sec, Cot};
(WRules = $TrigFns == (Through[$TrigFns[x]] /. x -> 2 ArcTan[t] // 
      TrigExpand // Together) // Thread)

Then, Partition[WRules /. Thread[$TrigFns -> Through[$TrigFns[x]]], 2] // TeXForm returns $$ \begin{align} \sin (x)&=\frac{2 t}{t^2+1}\,, & \cos (x)&=\frac{1-t^2}{t^2+1}\,, \\ \tan (x)&=-\frac{2 t}{t^2-1}\,, & \csc (x)&=\frac{t^2+1}{2 t}\,, \\ \sec (x)&=\frac{-t^2-1}{t^2-1}\,, & \cot (x)&=\frac{1-t^2}{2 t} \ . \end{align} $$

Then, we invert the rules using

invWRules = #[[1]] -> Solve[#, t, Reals] & /@ WRules

which we can finally use in the convert function:

convert[expr_, (trig : Alternatives@@$TrigFns)[x_]] := 
 Block[{temp, t},
  temp = expr /. x -> 2 ArcTan[t] // TrigExpand // Factor;    
  temp = temp /. (trig /. invWRules) // FullSimplify // Union;
  Or @@ temp /. trig -> HoldForm[trig][x] /. ConditionalExpression -> (#1 &)]

Note that the final line has HoldForm to prevent things like 1/Sin[x] automatically being rewritten as Csc[x], etc...

Here are some test cases - it is straight forward to check that the answers are correct (but don't forget to use RelaseHold):

In[6]:= convert[Sin[x], Cos[x]]
Out[6]= - Sqrt[1 - Cos[x]^2] || Sqrt[1 - Cos[x]^2]

In[7]:= convert[Sin[x]Cos[x], Tan[x]]
Out[7]= Tan[x]/(1 + Tan[x]^2)

In[8]:= convert[Sin[x]Cos[x], Cos[x]]
Out[8]= -Cos[x] Sqrt[1 - Cos[x]^2] || Cos[x] Sqrt[1 - Cos[x]^2]

In[9]:= convert[Sin[2x]Cos[x], Sin[x]]
Out[9]= -2 Sin[x] (-1 + Sin[x]^2)

In[10]:= convert[Sin[2x]Tan[x]^3, Cos[x]]
Out[10]= 2 (-2 + 1/Cos[x]^2 + Cos[x]^2)

---

A couple of quick thoughts about the above solution:

1. It assumes real arguments for the trig functions. It would be nice if it didn't do this and could be extended to hyperbolic trig and exponential functions.

2. When two solutions are given, it should return the domains of validity - or combine the appropriate terms using Abs[].

3. It should be extended to handle things like convert[Sin[x], Cos[2x]].

If anyone feels like implementing any of these things, please feel free!