The value of tanθ = ±∞ or tanθ = ±√(7/5)
The question is a trigonometric equation
<h3>What is a trigonometric equation?</h3>
A trigonometric equation is an equation that has unknowns that contain trigonometric ratios.
<h3>How to find the value of tanθ?</h3>
Given the trigonometric equation 7cosec²θ - 9cot²θ = 7, we require tanθ.
So, 7cosec²θ - 9cot²θ = 7
using the trigonometric identity 1 + cot²θ = cosec²θ.
Substituting this into the equation, we have
7cosec²θ - 9cot²θ = 7
7(1 + cot²θ)² - 9cot²θ = 7
7(1 + 2cot²θ + cot⁴θ) - 9cot²θ = 7
7 + 14cot²θ + 7cot⁴θ - 9cot²θ = 7
14cot²θ + 7cot⁴θ - 9cot²θ = 7 - 7
14cot²θ + 7cot⁴θ - 9cot²θ = 0
7cot⁴θ + 14cot²θ - 9cot²θ = 0
7cot⁴θ + 5cot²θ = 0
Factorizing out cot²θ, we have
cot²θ(7cot²θ - 5) = 0
⇒ cot²θ = 0 or 7cot²θ - 5 = 0
⇒ cotθ = ±√0 or 7cot²θ = 5
⇒ cotθ = ±0 or cot²θ = 5/7
⇒ cotθ = ±0 or cotθ = ±√(5/7)
⇒ 1/tanθ = ±0 or 1/tanθ = ±√(5/7)
⇒ tanθ = ±1/0 or tanθ = ±√(7/5)
⇒ tanθ = ±∞ or tanθ = ±√(7/5)
So, the value of tanθ = ±∞ or tanθ = ±√(7/5)
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