Question

please explain with proper steps ​

Answer 1

Answer:

(B) 2

Step-by-step explanation:

Let's make the algebra simpler by writing s = sin θ and c = cos θ.

Then the condition says

               s + s² = 1      ... (1)

and the expression to evaluate is

          c¹² + 3c¹⁰ + 3c⁸ + c⁶ + 1      ... (2)

Also, we know the identity

             c² + s² = 1.      ... (3)

From equation (3), we have c² = 1 - s², so from equation (1), c² = s.  Thus the expression in (2) is

  c¹² + 3c¹⁰ + 3c⁸ + c⁶ + 1

= s⁶ + 3s⁵ + 3s⁴ + s³ + 1

= s³ ( s³ + 3s² + 3s + 1 ) + 1

= s³ ( s + 1 )³ + 1

= ( s² + s )³ + 1

= 1³ + 1

= 1 + 1

= 2