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15 tan theta-5cot theta=0

$15 \tan \theta-5 \cot \theta=0$

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Solve equations involving like terms

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Q.

15 \tan \theta-5 \cot \theta=0

Understand equation and identities: Understand the given equation and identify the trigonometric identities involved. $\newline$ The given equation is $15 \tan(\theta) - 5 \cot(\theta) = 0$ . We know that $\tan(\theta)$ is the tangent of angle $\theta$ and $\cot(\theta)$ is the cotangent of angle $\theta$ . The cotangent is the reciprocal of the tangent, which means $\cot(\theta) = \frac{1}{\tan(\theta)}$ .
Express cot in terms: Express $\cot(\theta)$ in terms of $\tan(\theta)$ . $\newline$ Since $\cot(\theta) = \frac{1}{\tan(\theta)}$ , we can rewrite the equation as $15 \tan(\theta) - 5 \times \left(\frac{1}{\tan(\theta)}\right) = 0$ .
Find common denominator: Find a common denominator to combine the terms. $\newline$ The common denominator for $\tan(\theta)$ and $\frac{1}{\tan(\theta)}$ is $\tan(\theta)$ . Multiplying the second term by $\frac{\tan(\theta)}{\tan(\theta)}$ will give us a common denominator: $\newline$ $15 \tan(\theta) - 5 \times \left(\frac{\tan(\theta)}{\tan(\theta)}\right) \times \left(\frac{1}{\tan(\theta)}\right) = 0$ .
Simplify the equation: Simplify the equation. $\newline$ Now we have $15 \tan(\theta) - 5 \times (1) = 0$ , which simplifies to $15 \tan(\theta) - 5 = 0$ .
Add to isolate terms: Add $5$ to both sides of the equation to isolate terms with $\tan(\theta)$ . $\newline$ $15 \tan(\theta) - 5 + 5 = 0 + 5$ , which simplifies to $15 \tan(\theta) = 5$ .
Divide to solve for tan: Divide both sides of the equation by $15$ to solve for $\tan(\theta)$ . $\newline$ $\frac{15 \tan(\theta)}{15} = \frac{5}{15}$ , which simplifies to $\tan(\theta) = \frac{1}{3}$ .

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