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
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