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Evaluate

sin(pi+ theta) cos(pi/2 + theta) tan(3pi/2 - theta) cot(2pi- theta) / sin(2pi+theta) cos(2pi-theta) cosec (-theta) sin(3pi/2 + theta)

$\frac{\sin (π + θ) \cos (\frac{π}{2} + θ) \tan (\frac{3 π}{2} - θ) \cot (2 π - θ)}{\sin (2 π + θ) \cos (2 π - θ) \cos e c (- θ) \sin (\frac{3 π}{2} + θ)} = \frac{(- \sin θ) (- \sin θ) (\cot θ) (- \cot θ)}{(\sin θ) (\cos θ) (- \cos e c θ) (- \cos θ)} = \frac{- \sin^{2} θ \cot^{2} θ}{\sin θ (\cos^{2} θ) \times \frac{1}{\sin θ}} = \frac{- \sin^{2} θ \cot^{2} θ}{\cos^{2} θ} = - \tan^{2} θ \cot^{2} θ = - \tan^{2} θ \times \frac{1}{\tan^{2} θ} = - 1 H e n c e, \frac{\sin (π + θ) \cos (\frac{π}{2} + θ) \tan (\frac{3 π}{2} - θ) \cot (2 π - θ)}{\sin (2 π + θ) \cos (2 π - θ) \cos e c (- θ) \sin (\frac{3 π}{2} + θ)} = - 1$

-1

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