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Geometry Level 2

1 cos ⁡ 2 ( a ) + 1 1 + sin ⁡ 2 ( a ) + 2 1 + sin ⁡ 4 ( a ) + 4 1 + sin ⁡ 8 ( a ) \large \frac1{\cos^2(a)} + \frac1{1+\sin^2(a)} + \frac2{1+\sin^4(a) } +\frac4{1+\sin^8(a)}

If sin ⁡ 16 ( a ) = 1 5 \sin^{16} (a) = \frac15 , evaluate the expression above.


The answer is 10.

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M Rafi Adyatma by M Rafi Adyatma , 25, Indonesia

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2 solutions

M Rafi Adyatma
Jul 5, 2015

4 Helpful 1 Interesting 0 Brilliant 0 Confused

Moderator note:

Yes. This is just the repeated use of difference of squares: x 2 − y 2 = ( x + y ) ( x − y ) x^2- y^2= (x+y)(x-y) .

I'm curious whether there's any alternative solution.

Satyajit Mohanty
Jul 6, 2015

The expression turns to 8 1 − s i n 16 ( a ) \frac{8}{1 - sin^{16}(a)} Substitute the value provided to obtain the answer 10.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

It would be better to explain how the expression finally becomes 8 1 − sin ⁡ 16 ( a ) \frac8{1-\sin^{16}(a)} .

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