IIT JEE , AIEEE Forum - Mathematics , Trignometry

aug_crazy

6(cos⁶

+sin⁶

)-9(cos⁴

+sin⁴

)-3(cos

-sin

)²=?

spideyunlimited

Ans) 6 ( sin@cos@ - 1)

or

3 (sin2@ - 2)

bhuvana89

ya i too got the ans as 6(sinxcosx -1 )

Use this :

1. Sin⁶x + cos⁶x = 1-3sin²xcos²x

(u get this result by expanding ( sin²x ) ³ + ( cos²x )³

2. cos⁴x + sin⁴x = 1- 2 sin²xcos²x

use these to simplify and get the result

iitkgp_bipin

sin⁶x + cos⁶x = (sin²x)³ + (cos²x)³
= (sin²x + cos²x)(sin⁴x + cos⁴x - sin²x.cos²x)
= (1){(sin²x + cos²x)² - 2sin²x.cos²x - sin²x.cos²x}
= 1 - 3sin²x.cos²x

sin⁴x + cos⁴x = (sin²x + cos²x)² - 2sin²x.cos²x = 1 - 2sin²x.cos²x

Using these two :

6(cos⁶x + sin⁶x) - 9(cos⁴x + sin⁴x) - 3(cosx - sinx)²

= 6(1 - 3sin²x.cos²x) - 9(1 - 2sin²x.cos²x) - 3(cos²x+sin²x-2sinx.cosx)

= 6 - 18sin²x.cos²x - 9 + 18sin²x.cos² - 3(1 - 2sinx.cosx)

= 6(sinx.cosx - 1)

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