Show that :
(i) tan 48° tan 23° tan 42° tan 67° = 1
(ii) cos 38° cos 52° – sin 38° sin 52° = 0

(i) As we know that, tan 48° = tan (90° – 42°) = cot 42°

tan 23° = tan (90° – 67°) = cot 67°

So, we can rewrite the given expression as,

tan 48° tan 23° tan 42° tan 67° = tan (90° – 42°) tan (90° – 67°) tan 42° tan 67°

Substituting the obtained results,

 tan (90° – 42°) tan (90° – 67°) tan 42° tan 67° = cot 42° cot 67° tan 42° tan 67°

Club the required values, and we get,

cot 42° cot 67° tan 42° tan 67° = (cot 42° tan 42°) (cot 67° tan 67°) = 1×1 = 1

(ii) As we know that,

cos 38° = cos (90° – 52°) = sin 52°

cos 52°= cos (90°-38°) = sin 38°

So, we can rewrite the given expression as,

cos 38° cos 52° – sin 38° sin 52° = cos (90° – 52°) cos (90° – 38°) – sin 38° sin 52°

Substituting the obtained results,

cos (90° – 52°) cos (90° – 38°) – sin 38° sin 52° = sin 52° sin 38° – sin 38° sin 52° = 0