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When tackling expressions like \((a \sin x + b \cos x)^2 + (b \sin x - a \cos x)^2\), our first task is to expand the squares. This process involves applying the algebraic identity \((u + v)^2 = u^2 + 2uv + v^2\) for any terms \(u\) and \(v\).

For the term \((a \sin x + b \cos x)^2\), we identify \(u\) as \(a \sin x\) and \(v\) as \(b \cos x\).
Applying the identity, the expression becomes:

• \((a \sin x)^2\), which simplifies to \(a^2 \sin^2 x\).
• \(2(a \sin x)(b \cos x)\), which simplifies to \(2ab \sin x \cos x\).
• \((b \cos x)^2\), which simplifies to \(b^2 \cos^2 x\).

Together, they form \(a^2 \sin^2 x + 2ab \sin x \cos x + b^2 \cos^2 x\).

Now, for \((b \sin x - a \cos x)^2\), we use the same approach but with a small sign change in the middle term. This turns the mixed term into \(-2(b \sin x)(a \cos x) = -2ab \sin x \cos x\).
Thus, the expansion results in:

• \(b^2 \sin^2 x\)
• \(-2ab \sin x \cos x\)
• \(a^2 \cos^2 x\)

These pieces will help us simplify the expression further.

Trigonometric functions, such as \(\sin\) (sine) and \(\cos\) (cosine), play a critical role in trigonometry.We use them to relate angles to side lengths in right triangles.

In the context of our problem, \(\sin x\) and \(\cos x\) are the trigonometric identities applied within the expression.
To simplify or manipulate these functions, we often rely on certain fundamental identities:

• \(\sin^2 x + \cos^2 x = 1\): This is the Pythagorean identity, a pivotal tool in ensuring expressions simplify correctly.
• Angle addition formulas can also be beneficial, though less so for this specific problem.

In the exercise, understanding that the sine and cosine are both bounded functions between -1 and 1 is key.
They will scale the coefficients \(a\) and \(b\), but collectively will not alter the overall structure, meaning as we simplify, these identities will guide us.
Notice in the current expression how we consolidated terms based on trigonometric properties, simplifying tasks like everyday algebra.

Algebraic manipulation involves rearranging and simplifying expressions using basic algebraic rules.
In our scenario, after expanding the square expressions, the task becomes one of combining and simplifying terms.

To achieve this:

• Add terms like \(a^2 \sin^2 x\) from one expanded part to \(a^2 \cos^2 x\) from another.Through the identity \(\sin^2 x + \cos^2 x = 1\), this yields the term \(a^2\).
• Similarly, \(b^2 \sin^2 x + b^2 \cos^2 x = b^2\).
• The mixed terms \(2ab \sin x \cos x\) and \(-2ab \sin x \cos x\) conveniently cancel each other out.

Thus, the expression non-obviously simplifies to \(a^2 + b^2\).
Grasping these manipulations is crucial for effectively tackling various algebraic and trigonometric problems. They are fundamental skills for understanding more complex identities and transformations.