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sin x+tan x-(sin^(2)x)/(cos x)-1=0

$\sin x+\tan x-\frac{\sin ^{2} x}{\cos x}-1=0$

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Evaluate integers raised to rational exponents

Full solution

Q.

\sin x+\tan x-\frac{\sin ^{2} x}{\cos x}-1=0

Identify and Simplify: Identify and simplify the expression using trigonometric identities. $\newline$ $\sin x + \tan x - \frac{\sin^2 x}{\cos x} - 1 = 0$ $\newline$ $\tan x = \frac{\sin x}{\cos x}$ $\newline$ $\sin x + \frac{\sin x}{\cos x} - \frac{\sin^2 x}{\cos x} - 1 = 0$
Combine Like Terms: Combine like terms over a common denominator. $\frac{\sin x \cdot \cos x + \sin x - \sin^2 x}{\cos x - 1} = 0$
Simplify the Numerator: Simplify the numerator. $\frac{\sin x \cdot (\cos x + 1 - \sin x)}{\cos x} - 1 = 0$
Clear the Denominator: Multiply through by $\cos x$ to clear the denominator. $\newline$ $\sin x \cdot (\cos x + 1 - \sin x) - \cos x = 0$
Expand and Simplify: Expand and simplify the expression. $\sin x \cdot \cos x + \sin x - \sin^2 x - \cos x = 0$
Rearrange Terms: Rearrange terms to find a common factor. $\newline$ $(\sin x - \cos x) \cdot (\sin x + 1) = 0$
Solve Equations: Solve the equation by setting each factor to zero. $\newline$ $\sin x - \cos x = 0$ or $\sin x + 1 = 0$
Check Solutions: Solve each equation. $\newline$ $\sin x = \cos x$ $\newline$ $\sin x + 1 = 0$ $\newline$ $\sin x = -1$
Check Solutions: Solve each equation. $\newline$ $\sin x = \cos x$ $\newline$ $\sin x + 1 = 0$ $\newline$ $\sin x = -1$ Check the solutions in the original equation. $\newline$ $\sin x = \cos x$ , $x = \frac{\pi}{4}, \frac{5\pi}{4}$ $\newline$ $\sin x = -1$ , $x = \frac{3\pi}{2}$

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