Step 1: Compare the tan(a - b) expression with the given expression to identify the angles 'a' and 'b'.Take the example of tan(60º - 45º) to understand this better. Let us have a look at the below-given steps to learn the application of tan(a - b) identity. 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. Tan(a - b) = ST/TQ = cos a(tan a - tan b)/cos a(1 + tan a tan b) = (tan a - tan b)(1 + tan a tan b) ⇒ From right triangle UTQ, QT = cos a(QU) = cos a(QR + RU) = cos a(1 + tan a tan b) ⇒ From right triangle STP, ST = cos a(tan a - tan b)Įvaluating the linear pair formed at point S and applying the angle sum property of a triangle, we get, ∠RSU = a.Īlso, ∠PST = a Proof: Using trigonometric formulas on the right-angled triangle PRQ we get, Extend QR to point U and from this point, U, draw a perpendicular UT on PQ.
Take a point S on PR, such that ∠SQR = b, forming another right-angled triangle SRQ. tan b)Ĭonstruction: Assume a right-angled triangle PRQ with ∠PQR = a and base QR of unit length, as shown in the figure below.To prove: tan (a - b) = (tan a - tan b)/(1 + tan a But this formula, in general, is true for any value of a and b. In the geometrical proof of tan(a-b) formula, let us initially assume that 'a', 'b', and (a - b), i.e., (a > b). Let us see the stepwise derivation of the formula for the tangent trigonometric function of the difference of two angles.
We can give the proof of expansion of tan(a-b) formula using the geometrical construction method.
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