sec square theta+sec square theta(1-sec squared theta)-cosec square theta+cosec square theta (cosec square theta-1)=cot 4 theta-tan 4 theta

Asked by 2001anushkachoudhary | 28th Aug, 2016, 08:39: PM

Expert Answer:

begin mathsize 16px style To space prove space that colon
sec squared straight theta plus sec squared straight theta left parenthesis 1 minus sec squared straight theta right parenthesis minus cosec squared straight theta plus cosec squared straight theta left parenthesis cosec squared straight theta minus 1 right parenthesis equals cot to the power of 4 straight theta minus tan to the power of 4 straight theta
LHS space equals sec squared straight theta plus sec squared straight theta left parenthesis 1 minus sec squared straight theta right parenthesis minus cosec squared straight theta plus cosec squared straight theta left parenthesis cosec squared straight theta minus 1 right parenthesis
space space space space space space space space space equals sec squared straight theta plus sec squared straight theta minus sec to the power of 4 straight theta minus cosec squared straight theta plus cosec to the power of 4 straight theta minus cosec squared straight theta space
space space space space space space space space space equals 2 sec squared straight theta minus sec to the power of 4 straight theta minus 2 cosec squared straight theta plus cosec to the power of 4 straight theta space
space space space space space space space space space equals 2 sec squared straight theta minus open parentheses sec squared straight theta close parentheses squared minus 2 cosec squared straight theta plus open parentheses cosec squared straight theta close parentheses squared space
space space space space space space space space space equals 2 left parenthesis 1 plus tan squared straight theta right parenthesis minus left parenthesis 1 plus tan squared straight theta right parenthesis squared minus 2 left parenthesis 1 plus cot squared straight theta right parenthesis plus left parenthesis 1 plus cot squared straight theta right parenthesis squared.... left parenthesis Since space 1 plus tan squared straight theta equals sec squared straight theta space and space 1 plus cot squared straight theta equals cosec squared straight theta right parenthesis
space space space space space space space space space equals 2 plus 2 tan squared straight theta minus 1 minus 2 tan squared straight theta minus tan to the power of 4 straight theta minus 2 minus 2 cot squared straight theta plus 1 plus 2 cot squared straight theta plus cot to the power of 4 straight theta
space space space space space space space space space equals cot to the power of 4 straight theta minus tan to the power of 4 straight theta
space space space space space space space space space equals RHS
Hence space proved.

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Answered by Rebecca Fernandes | 28th Aug, 2016, 09:29: PM