We know that s e c squared theta - tan squared theta = 1.
When we substitute theta = 90 to the power of ring operator,
then, s e c squared90 to the power of ring operator - tan squared90 to the power of ring operator = infinity squared - infinity squared = 0.
Then how is it that the above identity is applicable for all possible values for theta, as is mentioned in many textbooks?

Asked by bjayanta | 22nd Jul, 2015, 09:54: PM

Expert Answer:

W h e n space theta equals straight pi over 2 comma space s e c straight pi over 2 equals infinity space a n d space tan straight pi over 2 equals infinity B o t h space s e c straight pi over 2 space a n d space t a n straight pi over 2 space a r e space v e r y space h i g h space v a l u e s space a n d space a r e space n o t space e q u a l space t o space e a c h space o t h e r. infinity space m e a n s space i n f i n i t e l y space a space l a r g e space v a l u e space a n d space i t space i s space j u s t space a space s y m b o l. S o space y o u space c a n n o t space s u b t r a c t space infinity space f r o m space infinity. T h e r e f o r e comma space t h e space s t a t e m e n t space infinity squared minus infinity squared equals 0 space i s space a space m e a n i n g space l e s s space s t a t e m e n t. A n d space h e n c e space y o u space n e e d space t o space f i n d space t h e space v a l u e space o f space s e c squared theta minus tan squared theta space w h e n space theta space t e n d s space t o space straight pi over 2 space a n d space n o t space w h e n space theta equals straight pi over 2. T h i s space c o n c e p t space i s space c a l l e d space c o n t i n u i t y space a n d space t h i s space w i l l space c o m e space i n space h i g h e r space c l a s s e s. C o n s i d e r space t h e space f o l l o w i n g space e x p l a n a t i o n colon stack l i m with theta rightwards arrow straight pi over 2 below open parentheses s e c squared theta minus tan squared theta close parentheses equals stack l i m with theta rightwards arrow straight pi over 2 below open parentheses fraction numerator 1 over denominator cos squared theta end fraction minus fraction numerator sin squared theta over denominator cos squared theta end fraction 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 equals stack l i m with theta rightwards arrow straight pi over 2 below open parentheses fraction numerator 1 minus s i n squared theta over denominator c o s squared theta end fraction 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 equals stack l i m with theta rightwards arrow straight pi over 2 below open parentheses fraction numerator c o s squared theta over denominator c o s squared theta end fraction close parentheses space space space space space space space space space space left square bracket because 1 minus s i n squared theta equals c o s squared theta right square bracket 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 stack l i m with theta rightwards arrow straight pi over 2 below open parentheses 1 close parentheses A s space t h e space l i m i t space d o e s space n o t space d e p e n d space u p o n space a n y space v a r i a b l e space a n d space 1 space i s space a space c o n s tan t comma space w e space h a v e comma stack l i m with theta rightwards arrow straight pi over 2 below open parentheses s e c squared theta minus t a n squared theta close parentheses equals 1 T o space s e r v e space y o u r space c u r i o s i t y space w e space a r e space p r o v i d i n g space t h e space s o l u t i o n. space Y o u space m a y space u n d e r s tan d space t h i s space c o n c e p t space w h e n y o u space c o m e space h i g h e r e r space c l a s s e s.  space space space space space space space space space space

Answered by Vimala Ramamurthy | 23rd Jul, 2015, 10:40: AM

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