1. tan x+ cot x=2 csc 2 x
sin(x)/cos(x)+cos(x)/sin(x)=(sin2(x)+cos2(x))/sin(x)cos(x).
sin2(x)+cos2(x)=1 (for all x); sin(2x)=2sin(x)cos(x), so sin(x)cos(x)=½sin(2x); therefore tan x+ cot x=2/sin(2x)=2csc(2x) QED.
2. cos ^4 x-sin^4x= cos2x
(cos2(x)+sin2(x))(cos2(x)-sin2(x));
cos(2x)=cos2(x)-sin2(x) (for all x),
therefore cos4(x)-sin4(x)=cos(2x) QED.
3. 1/2( 1-cos4 x)+ cos^2 2x=1
cos(4x)=2cos2(2x)-1 (for all x); 1-cos(4x)=2-2cos2(2x);
therefore ½(1-cos(4x))=1-cos2(2x) and
1-cos2(2x)+cos2(2x)=1 QED.
4. (sin 3x/sin x) -( cos 3x/cosx) =2
sin(3x)=sin(2x+x)=sin(2x)cos(x)+cos(2x)sin(x)=
2sin(x)cos2(x)+cos(2x)sin(x);
sin(3x)/sin(x)=2cos2(x)+cos(2x);
cos(3x)=cos(2x)cos(x)-sin(2x)sin(x)=
cos(2x)cos(x)-2sin2(x)cos(x);
cos(3x)/cos(x)=cos(2x)-2sin2(x);
sin(3x)/sin(x)-cos(3x)/cos(x)=
2cos2(x)+cos(2x)-(cos(2x)-2sin2(x))=
2(sin2(x)+cos2(x))=2 QED.
5. (sinx cos^2x )/1-sinx - sin x= sin^2x
sin(x)[cos2(x)/(1-sin(x))-1]=
sin(x)[cos2(x)-1+sin(x)]/(1-sin(x))=
sin(x)[-sin2(x)+sin(x)]/(1-sin(x))=
sin2(x)[-sin(x)+1]/(1-sin(x))=sin2(x)[1-sin(x)]/(1-sin(x))=sin2(x) QED.
6. (cos^2 x+sin x )/ 1-sin x= 1+ 2 sinx⇒
cos2(x)+sin(x)=(1-sin(x))(1+2sin(x))=1+sin(x)-2sin2(x)⇒
cos2(x)=1-2sin2(x) which is false, so the proposed identity is false.
However cos(2x)=1-2sin2(x) so:
(cos(2x)+sin(x))/(1-sin(x))=1+2sin(x) is a correct identity.
7. ( sec x-tan x) ^2= 2(1-sin x) ^2/ 1 +cos2x
1+cos(2x)=1+2cos2(x)-1=2cos2(x), so RHS=(1-sin(x))2/cos2(x);
sec(x)-tan(x)=1/cos(x)-sin(x)/cos(x)=(1-sin(x))/cos(x); therefore LHS=(1-sin(x))2/cos2(x)=RHS QED
8. -In (sec x-tanx) =In (sec x + tan x)⇒
ln(sec(x)+tan(x))+ln(sec(x)-(tan(x))=0⇒
ln(sec2(x)-tan2(x))=ln(1+tan2(x)-tan2(x))=ln(1)=0, therefore:
-ln(sec(x)-(tan(x))=ln(sec(x)+tan(x)) QED.
9. 2 sin x cos^2 (x/2) - (1/ 2) sin( 2x)= sin x
sin(2x)=2sin(x)cos(x) (for all x); sin(x)=2sin(x/2)cos(x/2), cos(x)=2cos2(x/2)-1 (for all x), therefore:
2sin(x)cos2(x/2)-sin(x)cos(x)=sin(x)(2cos2(x/2)-cos(x))=sin(x)(1+cos(x)-cos(x))=sin(x) QED.
10. (1-cosx)/sin x= sinx / (1 + cos x)⇒
(1-cos(x))(1+cos(x))=sin2(x), 1-cos2(x)=sin2(x) which is true for all x, therefore the proposed identity is true.
11. sin x -( cos x/ 1+ sin x)= tan x is not true for all x. For example, if x=0 this becomes:
0-1=0, -1=0 which is not true. However:
y-cos(x)/(1+sin(x))=y-cos(x)(1-sin(x))/(1-sin2(x))=
y-cos(x)(1-sin(x))/cos2(x)=y-(1-sin(x))/cos(x)=y-(sec(x)-tan(x)).
If y=sec(x) then y-cos(x)/(1+sin(x))=tan(x) and:
sec(x)-(cos(x)/(1+sin(x))=tan(x) is a true identity, true for all x.
12. 2 /( sin x + 1 )- 2 /( sin x - 1)= 4 sec ^2 x
2(sin(x)-1-(sin(x)+1)/(sin2(x)-1)=2(-2)/(-cos2(x))=4sec2(x) QED.
13. sec ^ 2 x + csc ^ 2 x = sec ^ 2x x csc ^ 2x
1/cos2(x)+1/sin2(x)=(sin2(x)+cos2(x))/(sin2(x)cos2(x))=
1/(sin2(x)cos2(x))=sec2(x)csc2(x) QED
14. (sec x +1) / tan x = tan x / (sec x -1)⇒
(sec(x)+1)(sec(x)-1)=tan2(x),
sec2(x)-1=tan2(x), or sec2(x)=1+tan2(x) which is true for all x and follows from:
sin2(x)+cos2(x)=1⇒sin2(x)/cos2(x)+1=sec2(x), that is, tan2(x)+1=sec2(x). So the proposition is true.