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Show that cotα – tanα = 2 cot 2α and hence deduce that tanα + 2 tan2α + 4 tan 4α+ 8 cot 8α = cot α

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Show that cotα – tanα = 2 cot 2α and hence deduce that tanα + 2 tan2α + 4 tan 4α+ 8 cot 8α = cot α

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cotα - tanα = 1/tanα - tanα

= (1-tan2α)/tanα

= 2.1/tan2α = 2cot2α

=>  tanα = cotα – 2 cot 2α 

= tanα + 2 tan2α + 4 tan 4α + 8 cot 8α 

= cotα – 2 cot 2α + 2(cot 2α – 2 cot 4α) + 4(cot 4α – 2 cot 8α) + 8 cot 8α 

= cotα

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