CBSE Class 10 Maths Trigonometry Question: If sin𝜃 + sin²𝜃 = 1, find the value of cos¹²𝜃 + 3cos¹⁰ 𝜃 + 3 cos⁸𝜃 + cos⁶𝜃 + 2 cos⁴𝜃 + 2 cos²𝜃 - 2 (#eduvictors)

Class 10 - Trigonometry Identity Based Question

Question: If sin𝜃 + sin²𝜃 = 1, find the value of 
        cos¹²𝜃 + 3cos¹⁰ 𝜃 + 3 cos⁸𝜃 + cos⁶𝜃 + 2 cos⁴𝜃 + 2 cos²𝜃 - 2 

Answer: 

Given,

sin𝜃 + sin²𝜃 = 1

sin𝜃 + sin²𝜃 = sin²𝜃 + cos²𝜃

∴ sin𝜃 = cos²𝜃

= cos¹²𝜃 + 3cos¹⁰ 𝜃 + 3 cos⁸𝜃 + cos⁶𝜃 + 2 cos⁴𝜃 + 2 cos²𝜃 - 2

= (cos¹²𝜃 + 3cos¹⁰ 𝜃 + 3 cos⁸𝜃 + cos⁶𝜃) + 2 (cos⁴𝜃 + cos²𝜃 - 1)

= [(cos⁴𝜃)³ + 3(cos⁴𝜃)²(cos²𝜃) + 3(cos⁴𝜃)(cos²𝜃)² + (cos²𝜃)³] + 2(sin²𝜃 + cos²𝜃 - 1)

= (cos⁴𝜃 + cos²𝜃)³ + 2 (1 - 1)

= (sin²𝜃 + cos²𝜃)³ + 2(0)

= (1)³ + 0

= 1