## 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

^{6}x - 48.cos

^{4}x + 18.cos

^{2}x - 6.cos

^{2}x - 1

## Proof

1st of all rewrite 3x as 3.2x

L.H.S. = cos 6x = cos (3.2x)

L.H.S. = cos 6x = cos (3.2x)

Now using the result cos 3Î¸ = 4cos

^{3}Î¸ - 3 cos Î¸ -----(1)

Replacing Î¸ as 2x in (1), we get

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

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

⇒ cos 2Î¸ = 2 cos2 Î¸ -1

L.H.S.= 4 {2cos

^{2}x -1}3 - 3 {2 cos

^{2}x -1}

**Now using the result {a - b }**

^{3}= {a}^{3}- { b }^{3}- 3{a }^{2}.b + 3(a). b^{2}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

Hence the Proof

##
Prove the Identity

##
tan (2x) = 2tan x 1 - tan^{2} 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

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 = cos^{2} x - sin^{2} x

##
**Proof**

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

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

cos 2x = cos

^{2}x - sin

^{2}x

^{ }

^{ }

^{}

Hence the Proof

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

Proof

Using the result

1+cos 2Î¸ = 2cos

^{2}Î¸
cos 2Î¸ = 2cos

^{2}Î¸ -1 -------------(1)
Replacing Î¸ with 2x in eq (1)

1+ cos 4x = 2cos

^{2}2x
cos 4x = 2cos

^{2}2x -1
Again using cos 2Î¸ = 2cos

^{2}Î¸ -1
cos 4x = 2(2cos

^{2}x -1)² -1It is the square of 2cos

^{2}x -1

cos 4x = 2(2cos

^{2}x -1)² -1
cos 4x = 2(4cos

^{4}x +1 - 4cos^{2}x) -1
cos 4x = 8cos

^{4}x +2 - 8cos^{2}x -1
cos 4x = 8cos

^{4}x - 8cos^{2}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

## Conclusion

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