Final answer:
The provided trigonometric statement appears to be incorrect as it does not align with the known trigonometric identities. For accurate proofs and solutions, properties and laws like the Pythagorean identity, sum and difference formulas, and the laws of sines and cosines should be applied.
Step-by-step explanation:
The statement ℓ sec (A-B) = \frac{\cos (A+B)}{\cos ^{2}(A) - \sin ^{2}(B)} can be proven using trigonometric identities. However, the formula provided seems to be incorrect as sec(θ) is defined as 1/\cos(θ) and the right side of the equation does not appear to simplify to a secant function. Normally, an identity involving secant would have an angle expressed as it is without addition or subtraction in its argument.
For trigonometric proofs, one often uses identities like the Pythagorean identity (\cos^2\theta + \sin^2\theta = 1), sum and difference formulas (\cos(A ± B), \sin(A ± B)), and properties of reciprocal trigonometric functions (\sec\theta = 1/\cos\theta, etc.) to transform and simplify the given expressions.
In the additional problems, such as problem #15 which states \sin a + \sin \beta = 2 \sin(a + \beta) \cos(a - \beta), we use sum-to-product formulas to combine or separate trigonometric functions.
Learning the law of sines and the law of cosines is crucial for solving problems involving triangles, especially in cases where classical right-triangle trigonometry is not applicable. These laws are used for finding unknown sides or angles in non-right triangles.