If tanβ=(n sinαcosα)/(1-n cos^2 α) then, tan(α+β) is equal to (a) (n-1) tanα(b) (n+1) tanα(c) (1

step 1 So we need to find the value of 10 alpha plus beta in this case. So as all option consists of 10

If tanβ=(n sinαcosα)/(1-n cos^2 α) then, tan(α+β) is equal...

If $\tan \beta=\frac{n \sin \alpha \cos \alpha}{1-n \cos ^{2} \alpha}$ then, $\tan (\alpha+\beta)$ is equal to
(a) $(n-1) \tan \alpha$
(b) $(n+1) \tan \alpha$
(c) $\frac{1}{n+1} \tan \alpha$
(d) $\frac{-1}{n-1} \tan \alpha$

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Key Concepts

Algebraic Manipulation in Trigonometry

Algebraic techniques such as finding a common denominator, factoring, and recognizing patterns are essential when working with trigonometric expressions. By correctly applying these methods along with known identities, one can simplify or solve trigonometric equations that might initially appear complex.

Pythagorean Identity

The fundamental trigonometric identity sin²? + cos²? = 1 underlies many simplification processes in trigonometry. It often allows the replacement of one trigonometric function with another, facilitating the reduction of expressions and the solving of equations.

Trigonometric Addition Formulas

These formulas, such as the tangent addition formula, enable the calculation of the trigonometric function of a sum (or difference) of angles in terms of the functions of the individual angles. In the context of the problem, the tangent addition formula helps combine tan ? and tan ? into a single expression, simplifying the task of finding tan(? + ?).

Tangent Function Properties

The tangent function, which is the ratio of sine to cosine, is central to many trigonometric identities and equations. Its properties, particularly how it behaves under angle addition or subtraction, are key to transforming complex expressions involving multiple trigonometric functions into simpler forms.

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