Evaluate : ` (cos^2(45 + theta) + cos^2(45 - theta))/(tan(60^@ + theta) tan(30^@ - theta)) + cosec(75^@ + theta) - sec(15^@ - theta)`

Explanation

LHS = `(cos^(2)(45^(@)+theta)+cos^(2)(45^(@)-theta))/(tan(60^(@)+theta).tan(30^(@)-theta))`
`(cos^(2)(45^(@)+theta)+ [sin{90^(@)-(45^(@)-theta)}]^(2))/(tan(60^(@)+theta).cot{90^(@)-(30^(@)-theta)})` `[therefore sin(90^(@)-theta)= costheta "and" cot(90^(@)-theta)= tan theta]`
`(cos^(2)(45^(@)+theta)+sin^(2)(45^(@)+theta))/(tan(60^(@)+theta).cot(60^(@)+theta))` `[therefore sin^(2)theta + cos^(2)theta=1]`
`1/(tan(60^(@)+theta). 1/(tan(60^(@)+theta)))=1="RHS" [therefore cottheta=1//tantheta]`