Formula, Proof, Examples

Tan(a - b) is one of the important trigonometric identities, also known as the tangent subtraction formula, used in trigonometry khổng lồ find the value of the tangent trigonometric function for the difference of angles. We can find the expansion of tan(a - b) lớn represent the chảy of a compound angle in terms of tangent trigonometric function for individual angles. Let us understand the expansion of tan(a-b) identity & its proof in detail in the following sections.

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1.	What is Tan(a - b) Identity in Trigonometry?
2.	Tan(a - b) Compound Angle Formula
3.	Proof of Tan(a - b) Identity
4.	Geometrical Proof of Tan(a - b) Formula
5.	How to Apply Tan(a - b)?
6.	FAQs on Tan(a - b)

What is Tan(a - b) Identity in Trigonometry?

Tan(a-b) identity is one of the trigonometry identities for compound angles. It is applied when the angle for which the value of the tangent function is khổng lồ be calculated is given in the form of the difference 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 lớn as the tangent subtraction formula in trigonometry. The tan(a-b) formula can be given as,

tan(a - b) = (tan a - tan b)/(1 + rã a·tan b)

Proof of Tan(a - b) Identity Using Sin (a - b) và Cos (a - b)

We can prove the expansion of tan(a - b) given as, tan(a - b) = (tan a - tan b)/(1 + chảy a·tan b) using the expansion of sin (a - b) và cos (a - b).

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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 & denominator by cos a cos b, we get

tan(a - b) = (tan a - chảy b)/(1 + rã 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 difference of two angles. In the geometrical proof of tan(a-b) formula, let us initially assume that 'a', 'b', & (a - b), i.e., (a > b). But this formula, in general, is true for any value of a & b.

To prove: tung (a - b) = (tan a - tan b)/(1 + tan a·tan b)

Construction: Assume a right-angled triangle PRQ with ∠PQR = a and base QR of unit length, as shown in the figure below. Take a point S on PR, such that ∠SQR = b, forming another right-angled triangle SRQ. Extend QR to lớn point U and from this point, U, draw a perpendicular UT on PQ.

Proof: Using trigonometric formulas on the right-angled triangle PRQ we get,tan a = PR/QR⇒ pr = QR chảy a⇒ pr = tung a (∵ QR = 1)

In right-triangle SRQ,tan b = SR/QR⇒ SR = QR rã b⇒ SR = tung b

⇒ PS = lăng xê - SR = rã a - tung b

⇒ From right triangle STP, ST = cos a(tan a - tung b)

Evaluating the linear pair formed at point S & applying the angle sum property of a triangle, we get, ∠RSU = a.Also, ∠PST = a

From right triangle URS,tan a = RU/SR⇒ RU = tan a rã b

⇒ From right triangle UTQ, QT = cos a(QU) = cos a(QR + RU) = cos a(1 + tung a tan b)

Finally, in right triangle STQ,

tan(a - b) = ST/TQ = cos a(tan a - tung b)/cos a(1 + rã a tan b) = (tan a - rã b)(1 + tan a tan b)

Hence, proved.

How to lớn 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 difference of standard angles in trigonometry. Let us have a look at the below-given steps to lớn learn the application of tan(a - b) identity. Take the example of tan(60º - 45º) khổng lồ understand this better.

Step 1: Compare the tan(a - b) expression with the given expression lớn identify the angles 'a' and 'b'. Here, a = 60º and b = 45º.Step 2: We know, tan(a - b) = (tan a - tan b)/(1 + chảy a·tan b)⇒ tan(60º - 45º) = (tan 60º - tan 45º)/(1 + chảy 60º·tan 45º)since, tung 60º = √3, rã 45º = 1⇒ tan(60º - 45º) = <√3 - 1>/<1 + (√3)·1> = (√3 - 1)/(√3 + 1).Also, we can compare this with the value of chảy 15º = (√3 - 1)/(√3 + 1). Therefore the result is verified.

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