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Question

Find tan3θ1+tan2θ+cot3θ1+cot2θsecθcscθ+2sinθcosθ

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Solution

tan3θ1+tan2θ+cot3θ1+cot2θsecθcosecθ+2sinθcosθ
1+tan2x=sec2x
1+cot2x=cosec2x
secx=1cosx
cosecx=1sinx
tanx=sinxcosx
cotx=cosxsinx
sin3θcos3θ×(sec2θ)+cos3θsin3θ×cosec2θ1cosθsinθ+2sinθcosθ
sin3θcosθ+cos3θsinθ1cosθsinθ+2sinθcosθ
sin4θ+cos4θ1+2sin2θcos2θsinθcosθ
(sin2θ)2+(cos2θ)2+2sin2θcos2θ1sinθcosθ
a2+b2+2ab=(a+b)2
(sin2θ+cos2θ)21sinθcosθ
sin2x+cos2x=1
121sinθcosθ=0

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