Please solve the following

Asked by manasd | 17th Sep, 2015, 10:34: PM

Expert Answer:

T h e space c o r r e c t space e q u a t i o n space i s P subscript n colon space t a n alpha plus 2 t a n 2 alpha plus 2 squared t a n 2 squared alpha plus... space u p space t o space apostrophe n apostrophe space t e r m s equals c o t alpha minus 2 to the power of n c o t open parentheses 2 to the power of n alpha close parentheses F o r space t h e space s t a t e m e n t space t o space b e space t r u e comma space f i r s t space w e space h a v e space t o space v e r i f y space i f space P subscript 1 space i s space t r u e. F o r space n equals 1 comma space R H S equals c o t alpha minus 2 c o t 2 alpha equals fraction numerator 1 over denominator tan alpha end fraction minus 2 fraction numerator 1 minus tan squared alpha over denominator 2 tan alpha end fraction equals fraction numerator 1 minus 1 plus t a n squared alpha over denominator tan alpha end fraction equals tan alpha equals L H S. H e n c e comma space P subscript 1 space i s space t r u e. space A s s u m e space P subscript n space i s space t r u e. rightwards double arrow t a n alpha plus 2 t a n 2 alpha plus 2 squared t a n 2 squared alpha plus... space u p space t o space apostrophe n apostrophe space t e r m s equals c o t alpha minus 2 to the power of n c o t open parentheses 2 to the power of n alpha close parentheses  P subscript n plus 1 end subscript colon L H S space equals space t a n alpha plus 2 t a n 2 alpha plus 2 squared t a n 2 squared alpha plus... space u p space t o space apostrophe n plus 1 apostrophe space t e r m s equals P subscript n plus t subscript n plus 1 end subscript equals c o t alpha minus 2 to the power of n c o t open parentheses 2 to the power of n alpha close parentheses plus 2 to the power of n tan 2 to the power of n alpha equals c o t alpha minus 2 to the power of n open parentheses fraction numerator 1 over denominator t a n 2 to the power of n alpha end fraction minus t a n 2 to the power of n alpha close parentheses equals c o t alpha minus 2 to the power of n open parentheses fraction numerator 1 over denominator t a n 2 to the power of n alpha end fraction minus t a n 2 to the power of n alpha close parentheses equals c o t alpha minus 2 to the power of n plus 1 end exponent open parentheses fraction numerator 1 minus tan squared 2 to the power of n alpha over denominator 2 t a n 2 to the power of n alpha end fraction close parentheses equals c o t alpha minus 2 to the power of n plus 1 end exponent c o t 2 open parentheses 2 to the power of n alpha close parentheses equals c o t alpha minus 2 to the power of n plus 1 end exponent c o t open parentheses 2 to the power of n plus 1 end exponent alpha close parentheses equals R H S H e n c e comma space P subscript n plus 1 end subscript space i s space t r u e space f o r space P subscript n space t o space b e space t r u e. space H e n c e comma space t h e space s t a t e m e n t space i s space a l w a y s space t r u e space b y space p r i n c i p l e space o f space m a t h e m a t i c a l space i n d u c t i o n.

Answered by satyajit samal | 18th Sep, 2015, 02:50: PM