@acer Thank you for your efforts in analyzing this situation in detail.

@Carl Love  and  @mmcdara  Thanks for the answers!

We get the following,

is(Or(cos(x) = -sqrt(1-sin(x)^2), cos(x) = sqrt(1-sin(x)^2))) assuming real;

  #  FAIL

It's a shame Maple doesn't know the right answer  true .

@mmcdara  The formula  cos(x)=(-4*sin(x/2)^4+sin(x)^2)/(4*sin(x/2)^4+sin(x)^2)
  may lead to errors for some argument values (i.e. these expressions are not equivalent). In addition, the expression becomes more complex. I still don't understand what the meaning of these transformations is?

eval(cos(x)=(-4*sin(x/2)^4+sin(x)^2)/(4*sin(x/2)^4+sin(x)^2), x=0);

       Error, numeric exception: division by zero

@Andiguys Yes, it is easy to do. Just remove x=10000 from your data. To plot 2 graphs for each vertical axis, use a separate list for each pair of graphs. Read the plot,details help for this.

 vv ,  mmcdara   Thank you for your interest and the solutions provided. My solution below differs from vv's solution in that I use  simplify  with siderals instead of  solve . Since  siderals  does not work with  abs, in order not to miss solutions I had to take into account different signs distributions (4 options in total):

restart;
A:=<x1,a>: B:=<x2,b>: C:=<x3,c>: vA:=<v1,0>: vB:=<v2,0>: vC:=<v3,0>:
f:=t->A+t*vA: g:=t->B+t*vB: h:=t->C+t*vC:
S:=t->1/2*LinearAlgebra:-Determinant(<g(t)-f(t) | h(t)-f(t)>):
P:=combinat:-permute([1,1,-1,-1],2):
{seq(simplify(abs(S(10)), {S(0)=2*p[1], S(5)=3*p[2]}), p=P)};

PS.  This problem is taken from a wonderful book (in Russian)  В.В. Прасолов "Задачи по планиметрии (часть 2)", which contains many original and non-trivial geometric problems on a variety of topics. This book can probably be found in translations into other languages.

If the number (-1) in the exponent simply means the reciprocal value, then the syntax is very simple

simplify(1/(diff(z, t)-diff(z, x)));

If you mean the inverse function, then clarification is required here, since you have a function of several variables.

@nilswe 

restart;
`&||` := (R::seq(algebraic)) -> `if`(_npassed = 0, infinity, 1/(add(`/`~(1, [R]))));
`&||`(R1,R2);

salim-barzani   Try   eq6 := eval(L, {x = Y/t, t^2 = Z, x^2 = X});

@JAMET  This is a completely different task and you should ask a separate question for it. Moreover, you must use English, because this is an English-language forum.

@JAMET   See my updated answer.

@Andiguys  Maybe so

P := evalf[4]([eval([k1, delta][], s[2]), s[1]]); 
display(plot3d(TRC(k1, delta), k1 = 0 .. 1, delta = 0 .. 1), pointplot3d(P, color = red, symbol = solidcircle, symbolsize = 17), textplot3d([P[], P], font = [times, 12], align = [ABOVE,RIGHT]), orientation = [45, 75], axes = normal);

@Andiguys  I didn't notice that the  0  after the delta should also be a subscript. The following option   labels=[ `δ`[0], s[2] ]    works for me.

@Earl  I think that for your purposes a procedure will be useful that will allow you to easily generate a chain of tangent circles of any length. The parameters of the  Circles  procedure are: n - circle's number, R - radius of the original circle, r1 - radius of the larger of the small circles. The procedure returns the center of the n-th small circle:

restart;
Circles:=proc(n,R,r1)
option remember;
if n=1 then return [2*sqrt(R*r1),r1] else
simplify([Circles(n-1,R,r1)[1]*sqrt(R)/(sqrt(R)+sqrt(Circles(n-1,R,r1)[2])),Circles(n-1,R,r1)[2]*R/(sqrt(R)+sqrt(Circles(n-1,R,r1)[2]))^2]) fi;
end proc:

# Examples of use

S0:=seq(Circles(n,4,1), n=1..15);
S:=seq(plottools:-circle(Circles(n,4,1),Circles(n,4,1)[2]), n=1..15):
plots:-display(plottools:-circle([0,4],4),S, scaling=constrained, view=[0..5.5,0..2.5],size=[900,600]);

@Rouben Rostamian  Thank you for this great construction! I converted your comment into an answer and vote up.

@Earl  Using the general formulas from my answer above, it is easy to verify that regardless of the size of the smaller circles, the centers of these circles are always on the parabola  y=x^2/(4*R) , where  R  is the radius of the original large circle. This is clearly visible in the animation:

restart;
OneFrame:=proc(R,r1)
local O,dist,O1,O20,O2,O3,O4,O5,O6,k1,k2,k3,k4,P1,P2,P3,P4,P5,P6:
uses plots, plottools;
dist:=(A,B)->sqrt((A[1]-B[1])^2+(A[2]-B[2])^2):
O:=[0,R]: O1:=[x1,y1]: O2:=[x2,y2]: O3:=[x3,y3]: 
O1:=eval(O1,solve({dist(O,O1)=R+r1,y1=r1}, {x1,y1},explicit)[1]):
O20:=eval([x2,y2],simplify(solve({dist(O1,O2)=r1+r2, dist(O,O2)=R+r2,y2=r2},{x2,r2,y2}, explicit)[1])):
O2:=[2*sqrt(R)*sqrt(r1)*sqrt(R)/(sqrt(R)+sqrt(r1)),r1*(sqrt(R)/(sqrt(R)+sqrt(r1)))^2]:
k1:=simplify(sqrt(R)/(sqrt(R)+sqrt(O2[2])));
O3:=simplify([O2[1]*k1,O2[2]*k1^2]);
k2:=simplify(sqrt(R)/(sqrt(R)+sqrt(O3[2])));
O4:=[O3[1]*k2,O3[2]*k2^2]:
k3:=simplify(sqrt(R)/(sqrt(R)+sqrt(O4[2])));
O5:=[O4[1]*k3,O4[2]*k3^2]:
k4:=simplify(sqrt(R)/(sqrt(R)+sqrt(O5[2])));
O6:=[O5[1]*k4,O5[2]*k4^2]:
P1:=circle(O1,O1[2],thickness=2); P2:=circle(O2,O2[2],thickness=2); P3:=circle(O3,O3[2],thickness=2); P4:=circle(O4,O4[2],thickness=2); P5:=circle(O5,O5[2],thickness=2); P6:=circle(O6,O6[2],thickness=2);

display(circle(O,R,thickness=2,color=blue),P1,P2,P3,P4,P5,P6,plot(x^2/(4*R), x=0..10, color=green, thickness=2), pointplot([O1,O2,O3,O4,O5,O6],color=red,symbol=solidcircle,symbolsize=4),scaling=constrained, view=[-4..12,0..9] );
end proc:


plots:-animate(OneFrame,[4,r1],r1=0.4..4,frames=73);