When you're working with tricky trigonometric equations in Grade 12, using angle sum and difference identities can really help make things easier. Here’s how I tackle it:

Understanding the Identities

First, it's important to know the basic sum and difference identities:

1. Angle Sum Identities:

   • For sine:
     (\sin(a + b) = \sin a \cos b + \cos a \sin b)
   • For cosine:
     (\cos(a + b) = \cos a \cos b - \sin a \sin b)
   • For tangent:
     (\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b})

2. Angle Difference Identities:

   • For sine:
     (\sin(a - b) = \sin a \cos b - \cos a \sin b)
   • For cosine:
     (\cos(a - b) = \cos a \cos b + \sin a \sin b)
   • For tangent:
     (\tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b})

Applying the Identities

When you face a complicated equation, here’s a simple way to solve it step by step:

1. Identify: Look for parts of the equation that can be rewritten as angle sums or differences. For example, if you see (\sin(15^\circ)), think of it as (\sin(45^\circ - 30^\circ)).

2. Substitute: Use the identities to replace those trigonometric terms. This can usually change the problem into something easier to handle.

3. Simplify: Combine similar terms and make the equation simpler. You might end up with a standard form that allows you to use known angles.

4. Solve: After simplifying, you can solve for an angle directly or use inverse trigonometric functions to find the angle values.

Practice Makes Perfect

Like anything in math, practice is really important. The more equations you work through using these identities, the more confident you’ll feel. Plus, it’s pretty satisfying to see a tough problem break down into something easy!