Prove that tan³theta÷1+tan²theta+cot³theta÷1+cot²theta=sec theta* cosec -2sintheta cos theta
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solution :
LHS = tan³ø/1 + tan²ø + cot³ø/1 + cot²ø
= tan³ø/sec²ø + cot³ø/cosec²ø
= sin³ø/cos³ø x cos²ø + cos³ø/sin³ø x sin²ø
= sin³ø/cos³ø + cos³ø/sinø
= sin^4ø + cos^4ø/sinø.cosø
= [sin²ø + cos²ø]² - 2 sin²ø.cos²ø/sinø.cosø
= (1)² - 2 sin²ø.cos²ø/sinø.cosø
= 1 - 2 sin²ø.cos²ø/sinø.cosø
= 1/sinø.cosø - 2 sin²ø.cos²ø/sinø.cosø
= secø. cosecø - 2 sinø. cosø
= RHS....... proved
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