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 θ - 3 cos θ  -----(1)

Replacing θ as 2x in (1), we get 

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

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

                                   ⇒ cos 2θ = 2 cos2 θ -1

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

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

Taking the product of powers to simplify it

cos 6x  =   4[ 8cos⁶ 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  = 32cos⁶ x  - 4 - 48cos⁴ x  + 24cos² x  - 6cos² x + 3

Adding the like powers terms and arranging in descending order

cos 6x   = 32cos⁶ x - 48cos⁴ 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 θ

cos 2θ = 2cos² θ -1 -------------(1)

Replacing θ with 2x in eq (1)

1+ cos 4x = 2cos² 2x

cos 4x = 2cos² 2x -1

Again using  cos 2θ = 2cos² θ -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 = 1- sin² 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

visit here for  EASY TRIGONOMETRY PART 1

and EASY TRIGONOMETRY PART 2

Conclusion

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