\( \left. \begin{array} { l } { \frac { 1 } { \sin x + 1 } - \frac { 2 } { \cos ^ { 2 } x } = \frac { \sin x + 1 } { \sin x - 1 } } \\ { \tan \theta + \frac { 1 } { \tan \theta } = \frac { \tan \theta } { \sin ^ { 2 } \theta } } \\ { \frac { \sin x } { ( 1 + \cos x ) } = \sqrt { \frac { 1 - \cos x } { 1 + \cos x } } } \\ { \sin ^ { 2 } \theta + \tan ^ { 2 } \theta ( \frac { \cos ^ { 2 } \theta } { \sin ^ { 2 } \theta } ) + \cos ^ { 2 } \theta = 2 } \\ { ( 2 \sin x - \cos x ) ( 2 \sin x + \cos x ) = 5 \sin ^ { 2 } x - 1 } \end{array} \right. \)

Let's dive into some fun facts about the world of trigonometry! Did you know that trigonometry originated in ancient civilizations, primarily the Greeks and Indians, who were using it for astronomical studies? The word "trigonometry" itself comes from the Greek words "trigonon" (triangle) and "metron" (measure). Over time, it has evolved into a vital field of mathematics used in various disciplines, from engineering to computer graphics! Now, if you're tackling tricky trigonometric equations, a common mistake is misapplying identities. Always double-check your fundamental identities, like \( \sin^2 x + \cos^2 x = 1 \) and the Pythagorean identity. Don’t forget to factor expressions correctly, as mistakes in sign can lead you astray. Keep your calculations neat and tidy, and you'll be solving these equations like a pro in no time!