Prove that sin(180° + θ)cos(90° + θ)tan(270° - θ)cot(360° - θ)/sin(360° - θ)cos(360° + θ)cosec(-θ)sin(270° + θ) = 1

Question

Prove that

\(\frac{sin(180^\circ\,+\,\theta)cos(90^\circ\,+\,\theta)tan(270^\circ-\,\theta)cot(360^\circ\,-\,\theta)}{sin(360^\circ-\,\theta)cos(360^\circ\,+\,\theta)cosec(-\theta)sin(270^\circ+\,\theta)} = 1\)

sin(180° + θ)cos(90° + θ)tan(270° - θ)cot(360° - θ)/sin(360° - θ)cos(360° + θ)cosec(-θ)sin(270° + θ) = 1

Answer 1

Using cos(90° + θ) = - sinθ(I quadrant cosx is positive

cosec( - θ) = - cosecθ

tan(270° - θ) = tan(180° + 90° - θ) = tan(90° - θ) = cotθ

(III quadrant tanx is positive)

Similarly sin(270° + θ) = - cosθ (IV quadrant sinx is negative

cot(360° - θ) = cotθ(IV quadrant cotx is negative)

= cotθ.tanθ.cotθ.tanθ \(\Rightarrow\) 1