(i) LHS = tan 5\(^\circ\) tan 25\(^\circ\)tan30\(^\circ\)tan 65\(^\circ\) tan 85\(^\circ\)= 1
tan (90\(^\circ\)- 85\(^\circ\)) tan( 90\(^\circ\)- 65\(^\circ\)) x \(\frac{1}{\sqrt3}\) x \(\cfrac{1}{cot60^\circ}\) \(\cfrac{1}{cot85^\circ}\)
cot 85\(^\circ\)cot 65\(^\circ\)\(\frac{1}{\sqrt3}\) \(\cfrac{1}{cot60^\circ}\) \(\cfrac{1}{cot85^\circ}\)
= \(\frac{1}{\sqrt3}\) = RHS
(ii) LHS = cot12\(^\circ\) cot 38\(^\circ\)cot 52\(^\circ\) cot 60\(^\circ\)cot78\(^\circ\)
= tan(90\(^\circ\) - 12\(^\circ\)) x tan (90\(^\circ\) - 38\(^\circ\)) x cot52\(^\circ\)x \(\frac{1}{\sqrt3}\)x cot 78\(^\circ\)
= \(\frac{1}{\sqrt3}\) x tan 78\(^\circ\)x tan 52\(^\circ\)x cot52\(^\circ\)x \(\cfrac{1}{tan52^\circ}\) \(\cfrac{1}{tan78^\circ}\)
= \(\frac{1}{\sqrt3}\)
= RHS
(iii) LHS = cos 15\(^\circ\) cos 35\(^\circ\) cosec55\(^\circ\) cos 60\(^\circ\)cosec 75\(^\circ\)
cos(90\(^\circ\)- 75\(^\circ\))cos (90\(^\circ\)- 55\(^\circ\))\(\cfrac{1}{sin55^\circ}\)x \(\frac{1}2\)x\(\cfrac{1}{sin75^\circ}\)
= sin75\(^\circ\)sin55\(^\circ\)\(\cfrac{1}{sin55^\circ}\) x \(\frac{1}2\)x\(\cfrac{1}{sin75^\circ}\)
= \(\frac{1}{2}\)
= RHS
(iv) LHS = cos1\(^\circ\) cos 2\(^\circ\) cos3\(^\circ\)......cos 180\(^\circ\)
= cos1\(^\circ\)x cos 2\(^\circ\)x cos3\(^\circ\)x......x cos 90\(^\circ\)x....x cos180\(^\circ\)
= cos1\(^\circ\)x cos 2\(^\circ\)x cos3\(^\circ\)x......x 0 x....x cos180\(^\circ\)
= 0
= RHS
(v) \((\cfrac{sin49^\circ}{cos41^\circ})\) + \((\cfrac{cos41^\circ}{sin49^\circ})\)= 2
= \((\cfrac{cos(90^\circ-49^\circ)}{cos41^\circ})^2\) + \((\cfrac{cos41^\circ}{cos(90^\circ-49^\circ)})^2\)
= \((\cfrac{cos41^\circ}{cos41^\circ})^2\) + \((\cfrac{cos41^\circ}{cos41^\circ})^2\)
= 12 + 12
= 1 + 1
= 2
= RHS
