i)Prove that:    square root of s e c theta minus 1 divided by s e c theta plus 1 end root space plus space square root of s e c theta plus 1 divided by s e c theta minus 1 end root space equals space 2 cos e c theta .
ii)Prove that: open parentheses 1 plus c o t squared theta close parentheses cross times open parentheses 1 plus cos theta close parentheses cross times open parentheses 1 minus cos theta close parentheses equals open parentheses 1 plus tan squared theta close parentheses cross times open parentheses 1 plus sin theta close parentheses cross times open parentheses 1 minus sin theta theta close parentheses equals 1

Asked by shreya ghoshal | 16th Oct, 2014, 06:13: AM

Expert Answer:

left parenthesis i right parenthesis square root of fraction numerator s e c theta minus 1 over denominator s e c theta plus 1 end fraction end root plus square root of fraction numerator s e c theta plus 1 over denominator s e c theta minus 1 end fraction end root equals fraction numerator square root of s e c theta minus 1 end root over denominator square root of s e c theta plus 1 end root end fraction plus fraction numerator square root of s e c theta plus 1 end root over denominator square root of s e c theta minus 1 end root end fraction equals fraction numerator open parentheses square root of s e c theta minus 1 end root close parentheses squared plus open parentheses square root of s e c theta plus 1 end root close parentheses squared over denominator square root of s e c theta plus 1 end root cross times square root of s e c theta minus 1 end root end fraction equals fraction numerator s e c theta minus 1 plus s e c theta plus 1 over denominator square root of s e c squared theta minus 1 end root end fraction equals fraction numerator 2 s e c theta over denominator tan theta end fraction equals 2 cross times fraction numerator 1 over denominator cos theta end fraction cross times fraction numerator cos theta over denominator sin theta end fraction equals fraction numerator 2 over denominator sin theta end fraction equals 2 cos e c theta
 
left parenthesis i i right parenthesis open parentheses 1 plus c o t squared theta close parentheses cross times open parentheses 1 plus cos theta close parentheses open parentheses 1 minus cos theta close parentheses equals open parentheses 1 plus c o t squared theta close parentheses cross times open parentheses 1 minus cos squared theta close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals open parentheses 1 plus c o t squared theta close parentheses cross times sin squared theta space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals cos e c squared theta cross times sin squared theta space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 1 over denominator sin squared theta end fraction cross times sin squared theta space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 H e n c e space f i r s t space p a r t space i s space p r o v e d. N o w space c o n s i d e r space open parentheses 1 plus tan squared theta close parentheses open parentheses 1 plus sin theta close parentheses open parentheses 1 minus sin theta close parentheses equals open parentheses 1 plus tan squared theta close parentheses open parentheses 1 minus sin squared theta close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals open parentheses 1 plus tan squared theta close parentheses cos squared theta space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals s e c squared theta cross times cos squared theta space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 1 over denominator cos squared theta end fraction cross times cos squared theta space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 H e n c e space t h e space s e c o n d space p a r t space i s space a l s o space p r o v e d.

Answered by Vimala Ramamurthy | 16th Oct, 2014, 09:14: AM