Prove that:
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Consider L.H.S.

= cosec² 58°- cot 58° tan 32° - tan 13° tan 37° tan 45° tan 53° tan 77°

= cosec² 58°- cot 58° tan (90-58)° - × [tan 13° tan 77°] × [tan 37° tan 53° ] × tan 45°

= cosec² 58°- cot 58° cot 58° - × [tan 13° tan (90-13)°] × [tan 37° tan (90-37)° ] × 1

= cosec² 58°- cot² 58° - × [tan 13° cot 13°] × [tan 37° cot 37° ]×1

= [cosec² 58°- cot² 58°] - × [1] × [1] × 1

= [1] -

= (2/3) – (5/3)

= (2 - 5)/3

= -3/3

= -1 = R.H.S.

Hence, proved.