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Question

Let E ( $alpha$ ) = $open square brackets table row cell cos to the power of 2 end exponent invisible function application alpha end cell cell cos invisible function application alpha sin invisible function application alpha end cell row cell cos invisible function application alpha sin invisible function application alpha end cell cell sin to the power of 2 end exponent invisible function application alpha end cell end table close square brackets$ . If $alpha$ and $beta$ differs by an odd multiple of $pi$ /2,then E( $alpha$ ) E( $beta$ ) is a -

null matrix
unit matrix
diagonal matrix
None of these

The correct answer is: null matrix

We have
E( $alpha$ )( $beta$ )= $open square brackets table row cell cos to the power of 2 end exponent invisible function application beta end cell cell cos invisible function application beta sin invisible function application beta end cell row cell cos invisible function application beta sin invisible function application beta end cell cell sin to the power of 2 end exponent invisible function application beta end cell end table close square brackets$
= $open square brackets table row cell cos invisible function application alpha cos invisible function application beta cos invisible function application blank left parenthesis alpha minus beta right parenthesis end cell cell cos invisible function application alpha sin invisible function application beta cos invisible function application left parenthesis alpha minus beta right parenthesis end cell row cell sin invisible function application alpha cos invisible function application beta cos invisible function application blank left parenthesis alpha minus beta right parenthesis end cell cell sin invisible function application alpha sin invisible function application beta cos invisible function application left parenthesis alpha minus beta right parenthesis end cell end table close square brackets$
As $alpha$ and $beta$ differ by an odd multiple of $pi$ /2, $alpha$ – $beta$ = (2n +1) $pi$ /2 for some integer n. Thus, cos [(2n +1) $pi$ /2] = 0
$therefore$ E( $alpha$ ) E( $beta$ ) = 0

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