If sinx+sin^2x=1 then with the value of cos^12 x+3cos^10x +3cos ^8x+cos^6x?

Question

Answer

Answer

Solution:

Step 1:

Using the identity sin^2x = 1 - cos^2x, substitute sin^2x in the given equation:

sinx (1 - cos^2x) = 1

sinx - sinx cos^2x = 1

sinx cos^2x = sinx - 1

Step 2:

Squaring both sides of the equation obtained in step 1:

sin^2x cos^4x = sin^2x - 2sinx + 1

Step 3:

Using the identity sin^2x = 1 - cos^2x, substitute sin^2x in the above equation:

(1 - cos^2x) cos^4x = 1 - 2sinx + sin^2x

cos^4x - cos^6x = 1 - 2sinx + (1 - cos^2x)

cos^4x - cos^6x = 2 - 2sinx - cos^2x

Step 4:

Multiplying both sides of the equation obtained in step 3 by cos^6x:

cos^10x - cos^12x = 2cos^6x - 2sinx cos^6x - cos^8x

Step 5:

Using the identity cos^2x = 1 - sin^2x, substitute cos^2x in the above equation:

cos^10x - cos^12x = 2cos^6x - 2sinx cos^6x - (1 - sin^2x)^4

Step 6:

Substituting sin^2x = 1 - cos^2x in the above equation:

cos^10x - cos^12x = 2cos^6x - 2sinx cos^6x - (1 - (1 - cos^2x))^4

cos^10x - cos^12x = 2cos^6x - 2sinx cos^6x - (cos^2x)^4

cos^10x - cos^12x = 2cos^6x - 2sinx cos^6x - cos^8x

Step 7:

Multiplying both sides of the equation obtained in step 6 by cos^2x:

cos^12x - cos^14x = 2cos^8x - 2sinx cos^8x - cos^10x

Step 8:

Substituting the value of cos^12

Answer