Tan(a + b)
Tan(a + b) is one of the important trigonometric identities, also known as tangent addition formulas, used in trigonometry to find the value of the tangent trigonometric function for the sum of angles. We can find the expansion of tan(a + b) to represent the tan of the given compound angle in terms of tangent trigonometric function for individual angles. Let us understand the expansion of tan(a+b) identity and its proof in detail in the following sections.
What is Tan(a + b) Identity in Trigonometry?
In trigonometry, tan(a+b) identity is one of the identities used for compound angles. It is applied when the angle for which the value of the tangent function is to be calculated is given in the form of the sum of any two angles. The angle (a+b) in the formula of tan(a+b) represents the compound angle.
Tan(a + b) Compound Angle Formula
Tan(a + b) formula for the compound angle (a+b) is referred to as the tangent addition formula in trigonometry. The tan(a+b) formula can be given as,
tan(a + b) = (tan a + tan b)/(1  tan a·tan b)
Proof of Tan(a + b) Identity Using Sin(a+b) and Cos(a+b)
We can prove the expansion of tan(a+b) given as, tan(a + b) = (tan a + tan b)/(1  tan a·tan b) using the expansion of sin(a+b) and cos(a+b).
we know, tan(a + b) = sin(a + b)/cos(a + b)
= (sin a cos b + cos a sin b)/(cos a cos b  sin a sin b)
Dividing the numerator and denominator by cos a cos b, we get
tan(a + b) = (tan a + tan b)/(1  tan a·tan b)
Hence, proved.
Geometrical Proof of Tan(a + b) Formula
We can give the proof of expansion of tan (a + b) formula using the geometrical construction method. Let us see the stepwise derivation of the formula for the tangent trigonometric function of the sum of two angles. In the geometrical proof of tan(a+b) formula, let us initially assume that 'a', 'b', and (a + b) are positive acute angles, i.e. (a + b) < 90. But this formula, in general, is true for any positive or negative value of a and b.
To prove: tan (a + b) = (tan a + tan b)/(1  tan a·tan b)
Construction: Assume a line segment RS of unit length, as shown in the figure below. Take a point T, such that RT forms an angle a with RS and ∠TSR = 90º, thus forming a rightangled triangle TSR. Taking TR as the base, construct a right angledtriangle UTR, with ∠URT = b and ∠UTR = 90°. Draw a line segment PQ with point U lying on it, such that PQ and RS form the two opposite sides of a rectangles PQSR, with point T lying on the side QS. Here, ∠URS = a + b < 90°.
Now, applying trigonometric formulas on the rightangled triangle TSR we get,
tan a = TS/RS
⇒ TS = 1 × tan a
⇒ TS = tan a
Also,
cos a = RS/RT
⇒ RT = 1/cos a
Similarly, in the right triangle UTR,
tan b = UT/RT
⇒ UT = RT × tan b
⇒ UT = tan b/cos a
We know, in right triangle UQT, since ∠UTQ = a (by equating the linear pair at point T and thus apply angle sum property of a triangle),
⇒ cos a = QT/UT
⇒ QT = (tan b/cos a) cos a
⇒ QT = tan b
Also, tan a = UQ/QT
⇒ UQ = tan a tan b
Using the calculated values, we conclude the measures of sides as,
RS = 1
PQ = 1
ST = tan a
QT = tan b
QS = ST + QT = tan a + tan b
UQ = tan a tan b
PU = PQ  UQ = 1  tan a tan b
PR = QS = tan a + tan b
Let us finally evaluate for right angled triangle UPR,
∠PUR = ∠URS = a + b [Alternate angles]
tan ∠PUR = PR/PU
tan(a + b) = (tan a + tan b)/(1  tan a·tan b)
Hence, proved.
How to Apply Tan(a + b)?
We can apply the expansion of tan(a + b) for finding the value of the tangent trigonometric function for angles that can be represented as the sum of standard angles in trigonometry. Let us have a look at the belowgiven steps to learn the application of tan(a + b) identity. Take the example of tan(30º + 45º) to understand this better.
 Step 1: Compare the tan(a + b) expression with the given expression to identify the angles 'a' and 'b'. Here, a = 30º and b = 45º.
 Step 2: We know, tan(a + b) = (tan a + tan b)/(1  tan a·tan b)
⇒ tan(30º + 45º) = (tan 30º + tan 45º)/(1  tan 30º·tan 45º)
since, tan 30º = 1/√3, tan 45º = 1
⇒ tan(30º + 45º) = [(1/√3) + 1]/[1  (1/√3)·1] = [(1 + √3)/√3]/[(√3  1)/√3] = (√3 + 1)/(√3  1).
Also, we can compare this with the value of tan 75º = (√3 + 1)/(√3  1). Therefore the result is verified.
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Let us have a look a few solved examples to understand tan(a+b) formula better.
Examples Using Tan(a + b)

Example 1: Find the exact value of tan 165º using expansion of tan(a + b).
Solution:
Since, the values of tan function can be easily calculated for 120º and 45º, we can write 165º as (120º + 45º).
⇒tan(165º) = tan(120º + 45º) = (tan 120º + tan 45º)/(1  tan 120º·tan 45º)º = [(√3) + 1]/[1  (√3)(1)] = (1  √3)/(1 + √3).

Example 2: Apply the tan(a + b) formula to find the expansion of the double angle formula tan 2θ.
Solution:
We can write tan 2θ = tan(θ + θ)
Applying tan(a + b) = (tan a + tan b)/(1  tan a·tan b)
tan 2θ = (tan θ + tan θ)/(1  tan θ·tan θ) = 2tan θ/(1  tan^{2}θ)
∴tan 2θ = 2tan θ/(1  tan^{2}θ)
FAQs on Tan (a + b)
What is Tan (a+b)?
Tan(a + b) is one of the important trigonometric identities also called tan addition formula in trigonometry. Tan (a + b) can be given as, tan (a + b) = (tan a + tan b)/(1  tan a·tan b), where 'a'and 'b' are angles.
What is the Formula of Tan(a+b)?
The tan (a + b) formula is used to express the tan compound angle formula in terms of tangent of individual angles. Tan (a + b) formula in trigonometry can be given as, (tan a + tan b)/(1  tan a·tan b).
What is Expansion of Tan(a+b)
The expansion of tan(a+b) is given as, tan (a + b) = (tan a + tan b)/(1  tan a·tan b). Here, a and b are the measures of angles.
How to Prove Tan(a+b) Formula?
The proof of tan(a+b) formula can be given using the geometrical construction method. We initially assume that 'a', 'b', and (a+b) are positive acute angles, i.e., (a + b) < 90. Click here to understand the stepwise method to derive tan(a + b) formula.
What are the Applications of Tan(a+b) Formula?
Tan(a+b) can be used to find the value of tangent function for angles that can be represented as the sum of standard or simpler angles. Thus, making the deduction easier. It can also be used in finding the expansion of other double and multiple angle trigonometry formulas.
How to Find the Value of Tan 15º Using Tan(a+b) Identity.
The value of tan 15º using (a + b) identity can be calculated by first writing it as tan[(45º+(30º)] and then applying tan(a+b) identity.
⇒tan[(45º+(30º)] = (tan 45º+ tan (30º))/(1  tan 45º·tan(30º)) = [1 + (1/√3]/[1  (1)(1/√3)] = [(√3  1)/√3]/[(√3 + 1)/√3] = (√3  1)/(√3 + 1).
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