Question

plz solve this:
It's urgent!!!!

Answer 1

Sin^2 A + Sin A - 1 = 0
sin A = [-1 + - √5]/2    or,   Sin A = (√5 - 1)/2
 
sin^2 A = (3-√5)/2
cos^2 A = sinA = (√5 -1)/2                    Cos^4 A = (3 - √5)/2
Cos^6 A = (√5-2)                                  cos^8 A = (7 - 3√5)/2
cos^10 A = (5√5 - 11)/2                        cos^12 A = (9 - 4√5)

Now find the value: by substituting these values:  answer = 1
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Another way:  without finding the values numerically

sin^2 A + sin A = 1
 => cos^2 A   = sin A
=>  cos^4 A   = sin^2 A = 1 - cos^2 A = 1 - sin A
=>  cos^6 A   = (1- sin A) sin A = sin A - 1 + cos^2 A = 2 sin A - 1
=>  cos^8 A   =  sin A (2 sin A -1) = 2 - 2 cos^2 A - sin A = 2 - 3 Sin A
=>  cos^10 A  = 2 sin A - 3 sin^2 A = 2 sin A - 3 + 3 cos^2 A = 5 sin A - 3
=>  cos^12 A  = sin A (5 sin A - 3) = 5 - 5 cos^2 A - 3 sin A = 5 - 8 sin A

substitute these in the given expression:
5 - 8 sin A + 3 (5 sin A - 3) + 3 (2 - 3 sin A) + 2 sinA -1 + 2(1-sin A) + 2 (sin A) - 2
= 1
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If the question is different like:
 sin(2*A)+sin(A)=1,  then   A = π/2.

cos 2A = -1.       Cos 4A = 1     Cos 6A = -1        cos 8A = 1      cos10A = -1
cos 12A = 1
substituting these values:  1 -3 + 3 -1 +2 - 2 -2 = -2

Answer 2

The answer is explained in the attachment below.


Hope it helps!