A solution of the equation (1-tanθ)(1+tanθ) sec^2 θ+2^tan^2 θ=0 where, θlies in interval (-(π)/(

step 1 1 minus tan theta into 1 plus tan theta into 6 square theta plus 2 raise 2 tan square theta equa

A solution of the equation (1-tanθ)(1+tanθ) sec^2 θ+2^tan^2...

A solution of the equation $(1-\tan \theta)(1+\tan \theta) \sec ^{2} \theta+2^{\tan ^{2} \theta}=0$ where, $\theta$ lies in interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ is given by $\theta$ is
(a) 0
(b) $\pm \frac{\pi}{3}$
(c) $-\frac{\pi}{6}$
(d) $\frac{\pi}{6}$

Key Concepts

Exponential Functions in Equations

Exponential functions, where the exponent itself is a variable or function of the variable, introduce unique challenges in solving equations. In the given problem, the term 2^(tan²?) necessitates understanding the behavior of exponential expressions and how they interact with polynomial terms to determine the overall solution.

Variable Substitution in Equation Solving

Substitution is used to transform a complex trigonometric equation into a simpler algebraic equation. By letting a new variable (such as u = tan²?) represent a trigonometric function, one can convert the original equation into an algebraic form involving polynomials and exponentials, making it easier to isolate and solve for the variable.

Pythagorean Trigonometric Identities

These identities, such as sec²? = 1 + tan²?, allow one to rewrite trigonometric expressions in equivalent forms. In this problem, using the identity transforms the given equation into a polynomial and exponential expression in terms of tan?, thereby simplifying the analysis and solution of the equation.

Algebraic Manipulation of Trigonometric Expressions

This concept involves rewriting products and sums of trigonometric functions—often by using known identities—to simplify complex equations. Here, recognizing that (1 - tan?)(1 + tan?) can be written as 1 - tan²? and then multiplied by sec²? helps reduce the equation to a form that is more tractable.

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