Geometry
# Fundamental Trigonometric Identities

\[ \large \frac { \sin ^{ 2 }{ \theta } }{ 5 } =\frac { \cos ^{ 2 }{ \theta } }{ 6 }\]

If \(\theta \) is a positive acute angle that satisfies the equation above, find \(\sin { \theta }\).

**Note:** Give your answer to 3 decimal places.

\[ \Large \left(\sqrt{2+\sqrt{2}}\right)^{x} + \left(\sqrt{2-\sqrt{2}}\right)^{x} = 2^{x}\]

Find the sum of all real \(x\) that satisfy the equation above.

\[ \large\frac { 1 }{ \cos ^{ 2 }{ \theta } } +\frac { 1 }{ 1+\sin ^{ 2 }{ \theta } } +\frac { 2 }{ 1+\sin ^{ 4 }{ \theta } } +\frac { 4 }{ 1+\sin ^{ 8 }{ \theta } } \]

If \( \large \sin ^{ 16 }{ \theta } = \frac { 1 }{ 5 } \), what is the value of the expression above?

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