Let `x=sin1^0,` then the value of the expression. `1/(cos0^0dotcos1^0)+1/(cos1^0dotcos2^0)+1/(cos2^0dotcos3^0)+1/(cos44^0dotcos45^0)` is equal to `x` (b) `1/x` (c) `(sqrt(2))/x` (d) `x/(sqrt(2))`

Explanation

`(2(sin1^@+sin2^@+sin3^@+...+sin89^@))/(2(cos1^@+cos2^@+cos3^@+...+cos44^@)+1)`

`=(2((sin1^@+sin89^@)+(sin2^@+sin88^@)+...+(sin44^@+sin46^@)+sin45^@))/(2(cos1^@+cos2^@+cos3^@+...+cos44^@)+1)`

`=(2(2sin45^@cos44^@)+(2sin45^@cos43^@)+...(2sin45^@cos1^@)+sin45^@)/(2(cos1^@+cos2^@+cos3^@+...+cos44^@)+1)`

`=(2*sin 45^@(2(cos1^@+cos2^@+cos3^@+...+cos44^@)+1))/(2(cos1^@+cos2^@+cos3^@+...+cos44^@)+1)`

`=2*sin45^@`

`=2(1/sqrt2)`

`=sqrt2`

so, option `(a)` is the correct option.