Question

If `cos(x) + cos^(2)x = 1`, then which of the following option is correct for, `sin^(12)x + 3sin^(10)x + 3sin^(8)x + sin^(6)x + 1`?

`A)0`
`B)1`
`C)2`
`D)3`

Answer 1

The answer is 2

Explanation:

Given that,
`cos x + cos^2x = 1 ... ... . ( 1 )`
`⇒ cos x = 1 − cos^2x`
`⇒ cos x = sin^2x`

When the given expression `sin^12x +3sin^10x + 3sin^8x +sin^6x+1`
`=(sin^2x)^6 +3sin^10x + 3(sin^2x)^4 +sin^6x+1`....from (1)

`= (cosx)^6 +3sin^10x + 3(cosx)^4+sin^6x+1`
`= cos^6x +sin^6x + 3cos^4x +3sin^10x +1`
`= (cos^2x +sin^2x)^3-3cos^2xsin^2x(cos^2x+sin^2x) + 3cos^4x +3(sin^2x)^5 +1`
`=1^3-3cos^2xcosx + 3cos^4x +3cos^5x+1`
`=1-3cos^3x(1- cosx) +3cos^5x +1`
`=1-3cos^3x*cos^2x +3cos^5x +1`
`=1-3cos^5x +3cos^5x +1`
`=1+1`
`=2`

Hope this helps.

Answer 2

`C)2`.

Explanation:

`cosx+cos^2x=1 rArr cosx=1-cos^2x=sin^2x......(star)`.

Now, `ul(sin^12x+3sin^10x+3sin^8x+sin^6x)+1`,

`=sin^6x{sin^6x+3sin^4x+3sin^2x+1}+1`,

`=(sin^2x)^3{(sin^2x+1)}^3+1`,

`={sin^2x(sin^2x+1)}^3+1`,

`={sin^4x+sin^2x}^3+1`,

`={(sin^2x)^2+sin^2x}^3+1`,

`={cos^2x+cosx}^3+1...........[because, (star)]`,

`=(1)^3+1................................................[because," Given]"`,

`=2,` as Respected Mohan V. has already derived!