46. \( (\sin x+\cos x)^{2}=1+\sin 2 x \)

Let's delve into a fun exploration of this equation! The expression \( (\sin x + \cos x)^2 \) can be expanded using the formula \( a^2 + b^2 + 2ab \), which transforms it into \( \sin^2 x + \cos^2 x + 2\sin x \cos x \). Since we know that \( \sin^2 x + \cos^2 x = 1 \), the equation simplifies beautifully to \( 1 + \sin 2x \), reinforcing the identity \( \sin 2x = 2\sin x \cos x \). It’s like a little party for trigonometric identities where everyone gets to know each other! Now, if you ever find yourself grappling with similar equations, remember that the key often lies in recognizing the fundamental identities. A common mistake is forgetting to apply \( \sin^2 x + \cos^2 x = 1 \), leading to messy equations. To avoid this, always look to simplify these expressions first before venturing into more complex territory. Keeping it clean can save you from getting tangled up in sin and cos confusion!