Question

IF cot theta + cosec theta = x, then find the value of tan theta in terms of x

Answer 1

Step-by-step explanation:

We have,

$\tt{cot (\theta) + cosec (\theta) = x}$

$\sf{ \implies cosec (\theta) = x - cot( \theta) }$

$\sf{ \implies cosec^{2} (\theta) = x^{2} + cot^{2} ( \theta) - 2 \cdot x \cdot cot (\theta)}$

$\sf{ \implies x^{2} + cot^{2} ( \theta) - cosec^{2} (\theta) - 2 \cdot x \cdot cot (\theta) = 0}$

$\sf{ \implies x^{2} -1 - 2 \cdot x \cdot cot (\theta) = 0}$

$\sf{ \implies x^{2} -1 = 2 \cdot x \cdot cot (\theta) }$

$\sf{ \implies cot (\theta) = \dfrac{ {x}^{2} - 1}{2x} }$

$\sf{ \implies tan(\theta) = \dfrac{2x}{ {x}^{2} - 1} }$