If sin theta sin2 theta 1 then what is the value of cos12 theta 3 cos10 theta 3 cos8 ...........

Quantitative Aptitude (CPO) Trignometry

If sin θ + sin² θ = 1, then what is the value of (cos¹² θ + 3 cos¹⁰ θ + 3 cos⁸ θ + cos⁶ θ - 1)?

-1

Let’s break it down step by step:

- Given: sin?? + sin²?? = 1

? Rearranged: sin?? + sin²?? – 1 = 0

? Factorization: (sin???–?1)(sin???+?1) = 0

? So, sin?? = 1 or sin?? = –1

- That means ? = 90° or ? = 270° (or their radian equivalents). For both, cos?? = 0.

- Now, look at the expression: cos¹²?? + 3cos¹°?? + 3cos8?? + cos6?? – 1

Plug in cos?? = 0 everywhere:

- cos6?? = 0

- cos8?? = 0

- cos¹°?? = 0

- cos¹²?? = 0

- Compute the expression:

0 + 0 + 0 + 0 – 1 = –1

So, the correct option is –1 (Option 1).

- Option 1: –1 (Correct value)

- Option 2: 0

- Option 3: 1

- Option 4: 2

Here’s the thing: The given values drop everything in that long expression except the –1 at the end. So, the answer has to be –1.

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