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If Aαβ=open square brackets table row cell cos invisible function application alpha end cell cell sin invisible function application alpha end cell row cell negative sin invisible function application alpha end cell cell cos invisible function application alpha end cell end table close square brackets, then which of following statement is true -

  1. A subscript alpha end subscript times A subscript beta end subscript equals A subscript alpha beta end subscript text  and  end text open parentheses A subscript alpha end subscript close parentheses to the power of n end exponent equals open square brackets table row cell cos to the power of n end exponent invisible function application alpha end cell cell sin to the power of n end exponent invisible function application alpha end cell row cell negative sin to the power of n end exponent invisible function application alpha end cell cell cos to the power of n end exponent invisible function application alpha end cell end table close square brackets    
  2. A subscript alpha end subscript times A subscript beta end subscript equals A subscript alpha beta end subscript text  and  end text open parentheses A subscript alpha end subscript close parentheses to the power of n end exponent equals open square brackets table row cell cos invisible function application n alpha end cell cell sin invisible function application n alpha end cell row cell negative sin invisible function application n alpha end cell cell cos invisible function application n alpha end cell end table close square brackets    
  3. A subscript alpha end subscript times A subscript beta end subscript equals A subscript alpha plus beta end subscript text  and  end text open parentheses A subscript alpha end subscript close parentheses to the power of n end exponent equals open square brackets table row cell cos to the power of n end exponent invisible function application alpha end cell cell sin to the power of n end exponent invisible function application alpha end cell row cell negative sin to the power of n end exponent invisible function application alpha end cell cell cos to the power of n end exponent invisible function application alpha end cell end table close square brackets    
  4. A subscript alpha end subscript times A subscript beta end subscript equals A subscript alpha plus beta end subscript text  and  end text open parentheses A subscript alpha end subscript close parentheses to the power of n end exponent equals open square brackets table row cell cos invisible function application n alpha end cell cell sin invisible function application n alpha end cell row cell negative sin invisible function application n alpha end cell cell cos invisible function application n alpha end cell end table close square brackets    

The correct answer is: A subscript alpha end subscript times A subscript beta end subscript equals A subscript alpha plus beta end subscript text  and  end text open parentheses A subscript alpha end subscript close parentheses to the power of n end exponent equals open square brackets table row cell cos invisible function application n alpha end cell cell sin invisible function application n alpha end cell row cell negative sin invisible function application n alpha end cell cell cos invisible function application n alpha end cell end table close square brackets


    To find the correct option.
    Given,  Aαβ=open square brackets table row cell cos invisible function application alpha end cell cell sin invisible function application alpha end cell row cell negative sin invisible function application alpha end cell cell cos invisible function application alpha end cell end table close square brackets
    A subscript alpha equals open square brackets table row cell cos space alpha end cell cell sin space alpha end cell row cell negative sin space alpha end cell cell cos space alpha end cell end table close square brackets

A subscript beta equals open square brackets table row cell cos space beta end cell cell sin space beta space end cell row cell negative sin space beta end cell cell cos space beta end cell end table close square brackets

A subscript alpha. A subscript beta equals open square brackets table row cell cos space alpha end cell cell sin space alpha end cell row cell negative sin space alpha end cell cell cos space alpha end cell end table close square brackets space x space equals open square brackets table row cell cos space beta end cell cell sin space beta end cell row cell negative sin space beta end cell cell cos space beta end cell end table close square brackets

space equals open square brackets table row cell cos space alpha plus beta space end cell cell sin space alpha plus beta end cell row cell negative sin space alpha plus beta end cell cell cos space alpha plus beta end cell end table close square brackets space equals space A subscript alpha plus beta end subscript

left parenthesis A subscript alpha right parenthesis to the power of n space equals space open square brackets table row cell cos space alpha end cell cell sin space alpha end cell row cell negative sin space alpha end cell cell cos space alpha end cell end table close square brackets to the power of n

l e t space n equals 2 comma

left parenthesis A subscript alpha right parenthesis squared space equals space open square brackets table row cell cos space alpha end cell cell sin space alpha end cell row cell negative sin space alpha end cell cell cos space alpha end cell end table close square brackets squared space equals space open square brackets table row cell cos space alpha end cell cell sin space alpha end cell row cell negative sin space alpha end cell cell cos space alpha end cell end table close square brackets space x space open square brackets table row cell cos space alpha end cell cell sin space alpha end cell row cell negative sin space alpha end cell cell cos space alpha end cell end table close square brackets

equals space open square brackets table row cell cos space 2 alpha end cell cell sin space 2 alpha end cell row cell negative sin space 2 alpha end cell cell cos space 2 alpha end cell end table close square brackets

T h e r e f o r e comma
left parenthesis A subscript alpha right parenthesis to the power of n space space equals space open square brackets table row cell cos space n alpha end cell cell sin space n alpha end cell row cell negative sin space n alpha end cell cell cos space n alpha end cell end table close square brackets

    A subscript alpha end subscript times A subscript beta end subscript equals A subscript alpha plus beta end subscript text  and  end text open parentheses A subscript alpha end subscript close parentheses to the power of n end exponent equals open square brackets table row cell cos invisible function application n alpha end cell cell sin invisible function application n alpha end cell row cell negative sin invisible function application n alpha end cell cell cos invisible function application n alpha end cell end table close square brackets is the correct option.

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