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

(i) tan 48° tan 23° tan 42° tan 67° Simplify the given problem by converting some of the tan functions to the cot functions We know that, tan 48° = tan (90° – 42°) = cot 42° tan 23° = tan (90° – 67°) = cot 67° = tan (90° – 42°) tan (90° – 67°) tan 42° tan 67° Substitute the values = cot 42° cot 67° tan 42° tan 67° = (cot 42° tan 42°) (cot 67° tan 67°) = 1×1 = 1 (ii) cos 38° cos 52° – sin 38° sin 52° Simplify the given problem by converting some of the cos functions to the sin functions We know that, cos 38° = cos (90° – 52°) = sin 52° cos 52°= cos (90°-38°) = sin 38° = cos (90° – 52°) cos (90°-38°) – sin 38° sin 52° Substitute the values = sin 52° sin 38° – sin 38° sin 52° = 0