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?