Show that cos (3π/2 + x) cos (2π + x) {cot (3π/2 − x) + cot (2π + x)} = 1.

Question

Show that cos \(\big(\frac{3π}{2}+x\big)\)cos (2π + x) {cot \(\big(\frac{3π}{2}-x\big)\) + cot (2π + x)} = 1.

Answer 1

 cos \(\big(\frac{3π}{2}+x\big)\)cos (2π + x) {cot \(\big(\frac{3π}{2}-x\big)\) + cot (2π + x)}

= cos (270° + \(x\)) cos (360° + \(x\)) {cot (270° – \(x\)) + cot (360° + \(x\))} 

= sin \(x\) cos \(x\) {tan \(x\) + cot \(x\)}

= sin \(x\) cos \(x\) \(\bigg[\frac{sin\,x}{cos\,x}+\frac{cos\,x}{sin\,x}\bigg]\) = sin \(x\) cos \(x\) \(\bigg[\frac{sin^2\,x+cos^2\,x}{sin\,x\,cos\,x}\bigg]\) = 1.