Trigonometric Identities – Sum and Difference & Sum-to-Product

Trigonometric Identities: Sum and Difference Formulas (with corrections and notes)

The transcript lists standard addition and subtraction formulas for sine, cosine, and tangent. Some entries contain typos (e.g., AB instead of A−B). The notes below present the correct forms and point out misprints observed in the transcript.

Sin addition:
\sin(A+B) = \sin A\cos B + \cos A\sin B
Cos addition:
\cos(A+B) = \cos A\cos B - \sin A\sin B
Tan addition:
\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A\tan B}
Sin subtraction (A − B):
\sin(A-B) = \sin A\cos B - \cos A\sin B
Cos subtraction (A − B):
\cos(A-B) = \cos A\cos B + \sin A\sin B
Tan subtraction (A − B):
\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A\tan B}
Transcript misprints and corrections:
- The line "sin(AB)=sin A cos B - cos A sin b" appears to intend sin(A−B). Correct form: \sin(A-B) = \sin A\cos B - \cos A\sin B
- The line "cos(AB) = cos A cos B + sin A sin B" appears to intend cos(A−B). Correct form: \cos(A-B) = \cos A\cos B + \sin A\sin B
- The line for tan(A−B) in the transcript is garbled; correct form is above: \tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A\tan B}
Important note on the structure of addition/subtraction identities:
- These identities are derived from the angle addition formulas and are fundamental in simplifying trigonometric expressions.
- They are consistent with the unit-circle definitions of sine and cosine and with the definitions of tangent as a ratio of sine and cosine.

Sum of sine with sum/difference:
\sin(A+B) + \sin(A-B) = 2\sin A\cos B
- This follows from adding the two expansion formulas for sin(A+B) and sin(A−B).
Product-like identity (often taught alongside):
\sin(A+B)\sin(A-B) = \sin^2 A - \sin^2 B
- Derivation sketch: let X = A+B and Y = A−B. Then sin X sin Y = (cos(X−Y) − cos(X+Y))/2 = (\cos(2B) - \cos(2A))/2, which simplifies to sin^2 A − sin^2 B when expressed via sine squares. The transcript’s version ("2 cos A sin B") is not correct for this product.
Sum of cosines with sum/difference:
\cos(A+B) + \cos(A-B) = 2\cos A\cos B
Difference of cosines with sum/difference:
\cos(A+B) - \cos(A-B) = -2\sin A\sin B
- This identity is a standard consequence of the cosine addition formula.
Transcript note about misprints here as well: the line "cos(A+B) cos(A-B) = -2 sin A sin B" is not correct for the product of cosines. The correct product-to-sum form is not equal to -2 sin A sin B; the proper identity is given above for the difference or via product-to-sum relations.

For any angles C and D:
- Sine addition:
  \sin C + \sin D = 2\sin\left(\frac{C+D}{2}\right)\cos\left(\frac{C-D}{2}\right)
- Sine subtraction:
  \sin C - \sin D = 2\cos\left(\frac{C+D}{2}\right)\sin\left(\frac{C-D}{2}\right)
- Cosine addition:
  \cos C + \cos D = 2\cos\left(\frac{C+D}{2}\right)\cos\left(\frac{C-D}{2}\right)
- Cosine subtraction:
  \cos C - \cos D = -2\sin\left(\frac{C+D}{2}\right)\sin\left(\frac{C-D}{2}\right)
Interpretations and uses:
- These identities are particularly useful for integrating products of sines and cosines or simplifying expressions where the angles are added or subtracted.
- They allow converting sums of trigonometric functions into products, which can simplify algebraic manipulation and integration tasks.

The transcript contains several typographical errors (e.g., AB instead of A−B, missing terms). Always verify with the standard sum/difference identities:
- \sin(A\pm B) = \sin A\cos B \pm \cos A\sin B
- \cos(A\pm B) = \cos A\cos B \mp \sin A\sin B
- \tan(A\pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}
Practical checks:
- For A = B, sin(A+B) becomes sin(2A) and tan(A+B) becomes tan(2A), which can be checked against double-angle formulas.
- Many of the sum-to-product identities reduce to simple forms when C = D, providing consistency checks.

Let A = 30°, B = 45°.
Using the addition formula:
\sin(30°+45°) = \sin 30°\cos 45° + \cos 30°\sin 45°
= \left(\tfrac{1}{2}\right)\left(\tfrac{\sqrt{2}}{2}\right) + \left(\tfrac{\sqrt{3}}{2}\right)\left(\tfrac{\sqrt{2}}{2}\right)
= \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} = \frac{\sqrt{2} + \sqrt{6}}{4} ≈ 0.9659
This matches sin(75°) as a consistency check.

Let A = 60°, B = 20°.
Compute both sides: left-hand side using the sum/difference formulas, right-hand side using the sum-to-product form.
Left-hand side (verify):
\sin(60°+20°) + \sin(60°-20°) = \sin(80°) + \sin(40°)
Numerically: sin(80°) ≈ 0.9848, sin(40°) ≈ 0.6428, sum ≈ 1.6276.
Right-hand side (2 sin A cos B):
2\sin(60°)\cos(20°) = 2\left(\tfrac{\sqrt{3}}{2}\right)\cos(20°)
cos(20°) ≈ 0.9397, so value ≈ 1.6276, confirming the identity.

These identities arise from the unit circle definitions of sine and cosine and from the geometric interpretation of angle addition in triangles and circular motion.
They are essential in solving trigonometric equations, simplifying integrals in calculus, and analyzing waves, signals, and rotations in physics and engineering.
Correct understanding and careful usage of these formulas prevent algebraic errors in applied problems (e.g., physics simulations, computer graphics rotations).

Addition/subtraction:
- \sin(A\pm B) = \sin A\cos B \;\pm\; \cos A\sin B
- \cos(A\pm B) = \cos A\cos B \;\mp\; \sin A\sin B
- \tan(A\pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}
Sum-to-product:
- \sin C + \sin D = 2\sin\left(\frac{C+D}{2}\right)\cos\left(\frac{C-D}{2}\right)
- \sin C - \sin D = 2\cos\left(\frac{C+D}{2}\right)\sin\left(\frac{C-D}{2}\right)
- \cos C + \cos D = 2\cos\left(\frac{C+D}{2}\right)\cos\left(\frac{C-D}{2}\right)
- \cos C - \cos D = -2\sin\left(\frac{C+D}{2}\right)\sin\left(\frac{C-D}{2}\right)
Product-to-sum (optional reminder, derived from above):
- \sin X\cos Y = \tfrac{1}{2}[\sin(X+Y) + \sin(X-Y)]
- \cos X\cos Y = \tfrac{1}{2}[\cos(X+Y) + \cos(X-Y)]
- \sin X\sin Y = \tfrac{1}{2}[\cos(X-Y) - \cos(X+Y)]$$
Note: Always cross-check transcript items with standard identities, since some lines in the provided transcript contain typographical errors (e.g., AB instead of A−B, and incorrect forms for some products).