Question

if cosA+cos²A=1
then prove that sin¹²A+3sin^10A+3sin^8A+sin^6A+2sin⁴A+2sin²A-2=1

Answer 1

cosA + cos²A = 1
cosA = 1 - cos²A = sin²A
cosA = sin²A use it here

here, LHS = sin^12A + 3sin^10A + 3sin^8A + sin^6A + 2sin^4A + 2sin^2A- 2

= (cosA)^6 + 3(cosA)^4.sin^2A + 3cos^2A.sin^4A + (sin^2A)^3 + 2(cos^2A)+2sin^2A - 2

= (cos^2A)^3 + 3(cos^2A)^2.sin^2A+3cos^2A.(sin^2A)^2 + 2(sin^2A + cos^2A) - 2

= (cos^2A + sin^2A)^3 + 2(sin^2A + cos^2A)-2

[ we know, sin²x + cos²x = 1 ]

= 1³ + 2.1² - 2 = 1 + 2 - 2 = 1 = RHS