Question

How do you use sum and difference identities to find the exact value of csc (-15)?

Answer 1

`csc(-15^@)=-sqrt2-sqrt6`

Explanation:

As cosecant ratio is inverse of sine ratio, we use here `sin(A-B)=sinAcosB-cosAsinB`

Let `A=30^@` and `B=45^@`

then `csc(-15^@)=1/sin(-15^@)=1/sin(30^@-45^@)`

= `1/(sin30^@cos45^@-cos30^@sin45^@)`

= `1/(1/2*1/sqrt2-sqrt3/2*1/sqrt2)`

= `(2sqrt2)/(1-sqrt3)`

= `(2sqrt2)/(1-sqrt3)*(1+sqrt3)/(1+sqrt3)`

= `(2sqrt2+2sqrt6)/(1^1-(sqrt3)^2)`

= `(2sqrt2+2sqrt6)/(1-3)`

= `-(2sqrt2+2sqrt6)/2`

= `-sqrt2-sqrt6`