Derive the identity for tan(α+β) using tan(α+β)=(sin(α+β))/(cos(α+β)) After applying the formula

Derive the identity for tan(α+β) using...

Key Concepts

Tangent Function Identity

The tangent function is defined as the ratio of the sine of an angle to its cosine. This relationship is pivotal when converting expressions involving sums of sines and cosines into an expression in terms of tangent, such as in the derivation of the tangent addition formula.

Sine Addition Formula

This formula expresses the sine of a sum of two angles as the sum of two sine and cosine products. It provides a method for breaking down a composite angle into its individual components, facilitating the derivation of other trigonometric identities, including the tangent addition formula.

Cosine Addition Formula

This formula gives the cosine of the sum of two angles as the difference of two cosine and sine products. It is crucial in the process of rewriting composite trigonometric functions in terms of their constituent angles, which is an essential step in deriving the tangent addition identity.

Algebraic Manipulation in Trigonometric Identities

Algebraic techniques such as dividing the numerator and denominator by a common factor are key tools in transforming complex trigonometric expressions. In the derivation of the tangent addition formula, dividing by the product of cosine values simplifies the expression and introduces the tangent function, making the identity more apparent.

Transcript

00:01 Hello students here we have to find out the formula for tan alpha plus beta using tan alpha plus beta is equal to sign alpha plus beta upon cost alpha plus beta.

00:16 So using this sign 10 alpha plus beta is equal to sine alpha plus beta upon cost alpha plus beta we'll have to find out the formula for 10 alpha plus beta so let's start so what we'll do here we know the formula for sine alpha plus beta and cost alpha plus beta so we have sign alpha plus beta this is of the form of sign a plus b right sign a plus b this comes out as sine a cos b sign a cos b plus cos a sine b so this is the formula for sine alpha a plus sign a plus b we'll use this formula and another formula we have here it is cost a plus b this comes out as cos a cos a cos b minus sine a sine a sine b so we'll use these two formulas and we'll find out the formula for 10 alpha plus beta so from this let's start here with 10 alpha plus beta 10 alpha plus beta here it is beta this comes out as we'll put the formula for sign alpha plus beta we'll get sign alpha sine beta and in denominator we have cost alpha plus beta so applying this formula again a is equal to alpha b is equal to beta so cos alpha cost alpha cos beta minus sine alpha sine alpha sine beta now what we'll do here we'll divide in numerator and denominator will divide by cos alpha cost beta we'll divide it we'll divide the numerator and denominator by cos alpha cos beta so let's see what we'll get so we have 10 alpha plus beta is equal to 10 alpha plus beta this comes out as we are dividing here by cos alpha cost alpha cost beta so first term will be sine alpha sign alpha cc beta upon cost alpha cos alpha cos beta so sign alpha cost beta upon cost alpha cost beta cost alpha cost plus next term will be it will be cons alpha, cos out of cost beta upon cos alpha, cos beta, sorry, kos alpha, sine beta it is college alpha sine beta upon cost alpha cause beta.

02:46 So cos alpha, here it is alpha sine beta upon cos alpha cost alpha cost beta, cos alpha, cost beta.

02:56 And we have to divide it in denominator also.

03:00 So here in denominator we have, first term is cost alpha, cost beta upon cost alpha cause beta, right? we are dividing by cost alpha, cos beta, so we'll get cost alpha, cost beta.

03:12 Cos beta upon cos alpha, it is cos alpha, cos beta...

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