String AB of unstretched length L is stretched by applying a force...

Refer the diagram applying Lami's thorem,

$\dfrac{F}{\sin \left(180 - 2\theta\right)} =$\dfrac{T}{\sin \left($90 +$\theta\right)}

$∴ \dfrac{F}{\sin \left(2\theta\right)}=\dfrac{T}{\cos \left(\theta\right)} ⇒$\dfrac{F}{2\sin \left(\theta\rright)\cos \left(\theta\right)}}=\dfrac{T}{\cos \left(\theta\right)}$</p><p>$∴ F = 2\sin \left(\theta\right) \timesT . . . \left(1\right)$</p><p>$butT = K \times∆ L = K \times \dfrac{L}{2} \times \left(\dfrac{1 - \cos \left(\theta\right)}{\cos \left(\theta\right)}\right)

put in equation~ \left(1\right),

$F = 2\sin \left(\theta\right) \timesK \times$\dfrac{L}{2} \times\left(\dfrac{1 - \cos \left(\theta\right)}{\cos \left(\theta\right)}\right)

∴ F = KL \left(1 - \cos \left(\theta\right)\right)\tan \left(\theta\right)