HOW TO PROVE TRIGONOMETRIC IDENTITIES || TRIGONOMETRY


Proof of trigonometric identities , trigonometric identities problems, proving trigonometric identities formulas,these trigonometric identities of class 10, fundamental trigonometric identities,trigonometric identities class 11 and its formation with the help of some examples.

How to prove Identity


cos 6x = 32cos6 x - 48.cos4 x   + 18.cos2 x  - 6.cos2 x  - 1

Proof

1st of all  rewrite 3x as 3.2x

L.H.S. = cos 6x =  cos (3.2x) 

Now using the result cos 3θ = 4cos3 Î¸ - 3 cos θ  -----(1)

Replacing Î¸ as 2x in (1), we get 

L.H.S. = 4cos3 2x - 3 cos 2x  -----------(2)


Now using the result  1+ cos 2θ = 2 cos2 Î¸ 

                                   ⇒ cos 2θ = 2 cos2 Î¸ -1

Replacing cos 2x = 2 cos2 x -1 in (2), we get 

L.H.S.= 4 {2cos2 x -1}3 - 3 {cos2 x  -1}


Now using the result {a - b }3 = {a}3 - b }3  -  3{a }2 .b   + 3(a). b2

  cos 6x    = 4[ {2cos² x  }3 - { 1 }3  -  3{2cos² x  }2 .1   +3.(2cos² x) .1² ] - 3 . {cos² x  -1}


Taking the product of powers to simplify it

cos 6x  =   4[ 8cos x  - 1 - 12cos⁴ x  + 6cos² x]  - 3{2cos² x-1}

Multiply by 4 in 1st term and multiply by -3 in 2nd term

 cos 6x  = 32cos x  - 4 - 48cos x  + 24cos² x  - 6cos² x + 3

Adding the like powers terms and arranging in descending order

cos 6x   = 32cos x - 48cos x  + 18cos² x  - 6cos² x  - 1

Hence the Proof



Prove the Identity 

tan (2x) =  2tan x  1 - tan2 x 

Proof

We know that 


tan (A+B) =  tan A +  tan B1 - tan A tan B 

Put A = B  = x in above formula . then it becomes

tan (x+x) =  tan x +  tan x1 - tan x tan x 


tan (2x) =  2tan x  1 - tan² x 
Hence the Proof


Prove that sin 2x = 2sin x cos x

Proof


As we know that sin (A + B) = sin A cos B + cos A sin B..  ...(1)

Put A = B  = x in ...   (1)

sin (x + x) = sin x cos x + cos x sin x

sin (2x) = sin x cos x +  sin x cos x

sin (2x) = 2 sin x cos x

Hence the Proof



Prove that cos 2x = cos2 x - sin2 x

Proof


As we know that cos (A + B) = cos A cos B - sin A sin B..  ...(1)
Put X = A = B in (1) , we get

cos (x + x) = cos x cos x - sin x sin x
cos 2x = cos2 x - sin2 x      


Hence the Proof

Prove that cos 4x = 8 cos⁴ x - 8 cos² x + 1


Proof    
 Using the result 
1+cos 2θ = 2cos2 Î¸
cos 2θ = 2cos2 Î¸ -1 -------------(1)
Replacing Î¸ with 2x in eq (1)
1+ cos 4x = 2cos2 2x
cos 4x = 2cos2 2x -1

Again using  cos 2θ = 2cos2 Î¸ -1

cos 4x = 2(2cos2 x -1)² -1

It is the square of 2cos2 x -1

cos 4x = 2(2cos2 x -1)² -1

cos 4x = 2(4cos4 x +1 - 4cos2 x) -1

cos 4x = 8cos4 x +2 - 8cos2 x -1

cos 4x =  8cos4 x - 8cos2 x +1

Hence the Proof


What is the value of sin3x?



To find the value of sin 3x ,  use this formula which contain sin (A+B)
therefore sin (A+B) = sin A cos B cos A sin B——-(1)
put A = 2x and B = x in (1)
then Sin 3x = sin 2x cos x + cos 2x sin x

As we know that cos 2x = 1 - 2sin³ x and sin 2x = 2 sin x cos x

Sin 3x = Sin (2x+x)
Sin 3x = sin 2x cos x + cos 2x sin x
sin 3x = (2 sin x cos x) cos x + (1 - 2sin³ x ) sin x
sin 3 x = 2 sin x cos² x + sin x -  2sin³ x

As we know that cos² x = 1sin² x

sin 3x= 2 sin x (1-sin³ x) + sin x - 2sin³ x
sin 3x = 2 sin x -2 sin³ x + sin x - 2sin³ x
sin 3x = 3 sin x - 4 sin³ x

Similarly we can prove that cos 3x= 4 cos³ x - 3 cos x
For learning and memorising more trigonometric formulas

Conclusion



In this post I have discussed trigonometric identities ,trigonometric identities problems, proving trigonometric identities formulas . If this post helped you little bit, then please share it with your friends to benefit them, comment your views on it to boost me and to do better, and also follow me on my Blog .We shell meet in next post till then Bye


If you are a mathematician Don't forget to visit my Mathematics You tube channel ,Mathematics Website and Mathematics Facebook Page , whose links are given below

Share:

4 comments:

  1. At this time, if you can use good medicinal materials and ingredients to strengthen hormones,Hsc Calculator will be able to retain your youth and enjoy masculine power.

    ReplyDelete
  2. This is one of the best blogs I've come across recently to learn maths. I appreciate the author's efforts in writing these educational articles. Find chemistry Edexcel Past Papers Maths.

    ReplyDelete
  3. I generally check this kind of article and I found your article which is related to my interest.Schools in Jaipur Genuinely it is good and instructive information. Thankful to you for sharing an article like this.

    ReplyDelete
  4. I generally check this kind of article and I found your article which is related to my interest. Genuinely it is good and instructive information. Thankful to you for sharing an article like this.RES Singapore

    ReplyDelete

Your valuable suggestions are always acceptable to us for betterment of this website