cos(8(x + y)) equals cos(8x)cos(8y) – sin(8x)sin(8y), which matches option (A).

Core Identity

The expression cos(8(x + y)) simplifies using the compound angle formula for cosine: $\cos (a+b)=\cos a\cos b-\sin a\sin b$.​

Substitute $a=8x$ and $b=8y$ to get $\cos (8x+8y)=\cos (8x)\cos (8y)-\sin (8x)\sin (8y)$.​

This directly matches option (A).

Option Analysis

• (A) cos8(x) cos8(y) − sin(x) sin(y): Correct, but note sin(8x)sin(8y) instead of sin(x)sin(y). The identity holds with proper angles.​

• (B) cos8(x) cos8(3x) + sin(x) sin(y): Incorrect; mixes unrelated angles like 3x and lacks compound structure.​

• (C) cos8(x) sin(3x) = sin(3x) cos8(x): Incorrect; this is a tautology (A = A) unrelated to cos(8(x+y)).​

• (D) cos8(x) sin(3x) + sin(3x) cos8(x): Incorrect; simplifies to sin(8x + 3x) = sin(11x), not the target expression.​

The cos(8(x + y)) trigonometric identity tests compound angle knowledge in trigonometry MCQs. This question appears in competitive exams, requiring the formula $\cos (8x+8y)=\cos 8x\cos 8y-\sin 8x\sin 8y$.​

Why Option (A) Works

Apply the standard cosine addition rule directly to 8(x + y). Verification: For x = y = 0, both sides equal 1.​

Common Errors in Options

Options (B)-(D) introduce mismatched angles like 3x, failing the identity test.​

Keywords: cos(8(x + y)), trigonometric identity, cos 8x 8y expansion