Question

Find the value of `(sin30^@-cos30^@)^2`?

Answer 1

`(sin30^@-cos30^@)^2=1-sin(2xx30^@)=1-sqrt3/2`

Explanation:

`(sinA-cosA)^2` is always equal to `1-sin2A`

The reason is `(sinA-cosA)^2`

= `sin^2A+cos^2A-2sinAcosA`

but as `sin^2A+cos^2A=1` and `2sinAcosA=sin2A`

it is `1-sin2A`

However, let us work it out for `A=30^@`

`(sin30^@-cos30^@)^2`

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

= `1/4+(sqrt3)^2/4-2xx1/2xxsqrt3/2`

= `1/4+3/4-sqrt3/2`

= `1-sqrt3/2`

and `1-sin2A=1-sin(2xx30^@)`

= `1-sin60^@`

= `1-sqrt3/2`

Hence `(sin30^@-cos30^@)^2=1-sin(2xx30^@)=1-sqrt3/2`