Proving an Identity

Question

 

 

1. tan x+ cot x=2 csc 2 x

2.  cos ^4 x-sin^4x= cos2x

3. 1/2( 1-cos4 x)+ cos^2 2x=1

4.  (sin 3x/sin x) -( cos 3x/cosx) =2

5. (sinx cos^2x )/1-sinx - sin x= sin^2x

6. (cos^2 x+sin x )/ 1-sin x= 1+ 2 sinx

7. ( sec x-tan x) ^2= 2(1-sin x) ^2/ 1 +cos2x

8. -In (sec x-tanx) =In (sec x + tan x)

9. 2 sin x cos^2 (x/2) - (1/ 2) sin( 2x)= sin x

10. (1-cosx)/sin x= sinx / (1 + cos x)

11. sin x -( cos x/ 1+ sin x)= tan x

12. 2 /( sin x + 1 )- 2 /( sin x - 1)= 4 sec ^2 x

13. sec ^ 2 x + csc ^ 2 x = sec ^ 2x x csc ^ 2x

14. (sec x +1) / tan x = tan x / (sec x -1)

 

 I needed your answer on any of the numbers. I found this in the book and until now, I only got 6  out  of 20 . And this 14  no. are the numbers I never prove. Thanks a lot for your answers. God bless.

 

 

Answer 1

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.