Trigonometric functions of sum and difference of two angles

The governing identity is the sine subtraction formula, sin(A − B) = sin A cos B − cos A sin B. Taking A = π and B = θ gives sin(π − θ) = sin π cos θ − cos π sin θ. Since sin π = 0 and cos π = −1, this becomes 0·cos θ − (−1)sin θ = sin θ. Therefore option B is correct. Geometrically, π − θ and θ have equal heights on the unit circle, so their sine values are equal; they are reflections across the vertical line through π/2. Option A introduces an incorrect sign change, while options C and D confuse sine with cosine. The identity is valid for every real θ, not just for acute angles.

Apply the cosine subtraction identity, cos(A − B) = cos A cos B + sin A sin B. With A = π and B = θ, cos(π − θ) = cos π cos θ + sin π sin θ. Since cos π = −1 and sin π = 0, the value is (−1)cos θ + 0·sin θ = −cos θ. Thus option C is correct. On the unit circle, the angle π − θ lies in the reflected position whose y-coordinate agrees with that of θ but whose x-coordinate is opposite. Because cosine represents the x-coordinate, its sign changes. Option A fails to include the negative sign, while options B and D use sine rather than cosine and therefore represent the wrong coordinate.

The tangent subtraction identity is \(\tan(A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B}\), where defined. More simply, tangent has period \(\pi\) and is odd under sign reversal: \(\tan(\pi- heta)=\tan(-\theta)=-\tan\theta\). Hence option D is correct. The same result follows from quadrant signs: the angle \(\pi- heta\) lies in the second quadrant when \(0<\theta<\pi\), where sine is positive and cosine is negative, so their ratio is negative. Option A ignores the sign change; B and C replace tangent by its reciprocal, cotangent.

The governing concept is the sine difference identity. Use sin(a − b) = sin a cos b − cos a sin b with a = θ and b = π. Because cos π = −1 and sin π = 0, sin(θ − π) = sin θ(−1) − cos θ(0) = −sin θ. Thus option B is correct. This can also be understood through periodicity: θ − π and θ + π differ by 2π, so their sine values are equal, and the sine of either angle is −sin θ. Option A misses the sign reversal caused by a half-turn. Options C and D use cosine instead of sine, so they do not represent the required coordinate or function. The identity applies wherever the trigonometric expression is defined.