Question

Verify the identity. \(\tan A+\tan B=\frac{\sin (A+B)}{\cos A \cos B}\)

Short answer

We can verify the given trigonometric identity by rewriting it in terms of sine and cosine, finding a common denominator, applying the sine of the sum formula, and substituting it into the left-hand side. This results in the same expression as the right-hand side, confirming the identity: \[\tan A + \tan B = \frac{\sin(A+B)}{\cos A \cos B}.\]

Step-by-step solution

Rewrite the expressions in terms of sine and cosine

Recall that $$\tan x = \frac{\sin x}{\cos x}.$$ So we can rewrite the given expression as: $$\frac{\sin A}{\cos A} + \frac{\sin B}{\cos B} = \frac{\sin (A+B)}{\cos A \cos B}.$$

Find a common denominator for the left-hand side (LHS)term

Since we are dealing with fractions on the LHS, we need to find a common denominator for both terms. The common denominator here is \(\cos A \cos B\), so we can rewrite the LHS as $$\frac{\sin A \cos B + \sin B \cos A}{\cos A \cos B}.$$

Apply the sine of the sum formula

Recall the sine of the sum formula: $$\sin (A+B) = \sin A \cos B + \cos A \sin B.$$ Comparing this formula with the numerator of the LHS, it is clear that $$\sin (A+B) = \sin A \cos B + \sin B \cos A.$$

Substitute the sine of the sum formula into the left-hand side and verify identity

We can now substitute the sine of the sum formula into the numerator of the LHS to get $$\frac{\sin (A+B)}{\cos A \cos B}.$$ Now, by comparing this result with the right-hand side (RHS) of the given identity, we can see that they are the same. Therefore, the given identity is verified: $$\tan A + \tan B = \frac{\sin (A+B)}{\cos A \cos B}.$$

Key concepts

Sine and Cosine

Understanding sine and cosine functions is fundamental to trigonometry. These functions relate the sides of a right triangle to its angles. The sine of an angle, often denoted as \(\sin\), is the ratio of the length of the side opposite the angle to the length of the hypotenuse. In formula terms, if you have a right triangle with an angle \( A \) and sides \( a \) (opposite to \( A \) ), \( b \) (adjacent to \( A \) ), and \( c \) (the hypotenuse), \( \sin(A) = \frac{a}{c} \).

The cosine of an angle, denoted as \(\cos\), is the ratio of the length of the adjacent side to the length of the hypotenuse. Using the same triangle, the formula would be \( \cos(A) = \frac{b}{c} \). Both sine and cosine values are essential in calculating various trigonometric identities, including the tangent function, which is the ratio of sine to cosine.

Sum of Angles Formula

The sum of angles formula is a pivotal concept within trigonometry, especially when dealing with sine and cosine functions. This formula allows the calculation of the sine (or cosine) of the sum of two angles when the sines and cosines of these two angles are known. For angles \( A \) and \( B \) the formula for sine is \( \sin(A+B) = \sin(A)\cos(B) + \cos(A)\sin(B) \).

The importance of this formula is highlighted when simplifying trigonometric expressions or verifying identities. The ability to break down the sine of a sum into its components is a powerful tool that can be used to demonstrate the equivalence of seemingly different trigonometric expressions, as seen in the given exercise.

Verifying Trigonometric Identities

Verifying trigonometric identities involves proving that two trigonometric expressions are equivalent, regardless of the value of the variables involved. To do this, one typically manipulates one or both sides of the equation using known identities, such as the sum of angles formula, until both sides have the same form. A trusted approach includes expressing all trigonometric terms in sine and cosine, because they serve as the foundation to other trigonometric functions.

Key strategies in verification include finding common denominators for fractions, using reciprocal identities, like \( \cot\) being the reciprocal of \( \tan \), and angle sum or difference formulas. In the provided exercise, the sine sum formula is used to verify the identity, showing that both the left-hand side (LHS) and the right-hand side (RHS) represent the same value.

Common Denominators in Algebra

Common denominators are a crucial concept in algebra, especially when dealing with fractions. To combine fractions, they must have the same denominator. Finding a common denominator involves determining the least common multiple (LCM) of the denominators. In the context of trigonometry, combining terms with different trigonometric functions (each having distinct denominators like \( \cos(A) \) and \( \cos(B) \)) involves finding a common base, which usually results in a product of the distinct denominators.

In the exercise, the common denominator for the LHS was \( \cos(A)\cos(B) \), which facilitated the combination of the two terms into a single fraction. This step is essential for verifying trigonometric identities because it simplifies the equation, making it easier to see the relationship between the two sides and confirm their equivalence.