If cos(A-B)=(3)/(5) and tanA tanB=2, then (a) cosA cosB=(1)/(5) (b) sinA sinB=(-2)/(5) (c) cos(A

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If cos(A-B)=(3)/(5) and tanA tanB=2, then (a) cosA...

If $\cos (\mathrm{A}-\mathrm{B})=\frac{3}{5}$ and $\tan \mathrm{A} \tan \mathrm{B}=2$, then
(a) $\cos A \cos B=\frac{1}{5}$
(b) $\sin \mathrm{A} \sin \mathrm{B}=\frac{-2}{5}$
(c) $\cos (\mathrm{A}+\mathrm{B})=\frac{-1}{5}$
(d) $\sin \mathrm{A} \sin \mathrm{B}=\frac{4}{5}$

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Key Concepts

Cosine Addition/Subtraction Formulas

These formulas, such as cos(A ? B) = cos A cos B + sin A sin B and cos(A + B) = cos A cos B ? sin A sin B, express the cosine of sums or differences of angles in terms of the individual sines and cosines. They are fundamental in solving trigonometric equations, simplifying expressions, and establishing relationships between the trigonometric functions of multiple angles.

Relationship between Tangent Product and Sine-Cosine Expressions

Since the tangent function can be represented as the ratio of sine to cosine (tan A = sin A / cos A), the product tan A tan B can be rewritten as (sin A sin B)/(cos A cos B). This relationship connects the product of tangents to the products of sines and cosines, allowing one to relate and solve equations by expressing sine and cosine products in terms of known tangent values.

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