Trigonometry> Sin theta + sin^2theta + sin ^3theta = 1 ...

Sin theta + sin^2theta + sin ^3theta = 1 prove that cos^6theta - 4cos^4theta + 8cos^2theta=4

Anurag patel , 6 Years ago

Grade 10

2 Answers

Aditya Gupta

hello anurag,

Given relation sin x + sin 2 x + sin 3 x = 1 ⇒ sin x + sin 3 x = 1 − sin 2 x , now squaring ⇒ ( sin x + sin 3 x ) 2 = ( 1 − sin 2 x ) 2 ⇒ sin 2 x + sin 6 x + 2 sin 4 x = cos 4 x ⇒ 1 − cos 2 x + ( 1 − cos 2 x ) 3 + 2 ( 1 − cos 2 x ) 2 = cos 4 x ⇒ 1 − cos 2 x + 1 − 3 cos 2 x + 3 cos 4 x − cos 6 x + 2 − 4 cos 2 x + 2 cos 4 x = cos 4 x ⇒ cos 6 x − 4 cos 4 x + 8 cos 2 x = 4

kindly approve :)

Last Activity: 6 Years ago

Vikas TU

Dear student

Here is the correct explanation

sinA + sin^2A + sin^3A = 1
=> sinA + sin^3A = 1 - sin^2A
=> sinA ( 1+sin^2A ) = cos^2A
=> sinA ( 1 + 1 - cos^2A ) = cos^2A
=> sinA ( 2 - cos^2A ) = cos^2A
=> sinA = cos^2A / ( 2 - cos^2A)
=> sqrt ( 1 - cos^2A ) = cos^2A / 2- cos^2A
=> 1 - cos^2A = ( cos^2A )2 / ( 2 - cos^2A )2
=> 1 - cos^2A = cos^4A / ( 4 + cos^4A - 4 cos^2A )
=> ( 1 - cos^2A ) ( 4 + cos^4A - 4 cos^2A ) = cos^4A
=> ( 4 + cos^4A - 4 cos^2A -4cos^2A - cos^6A + 4cos^4A = cos^4A
=> ( 4 -8cos^2A - cos^6A + 4cos^4A ) = cos^4A - cos^4A
=> ( 4 -8cos^2A - cos^6A + 4 cos^4A ) = 0
=> 4 = 8cos^2A + cos^6A - 4cos^4A

Last Activity: 6 Years ago