Question

if `cos theta = 5/13` find the value of `(sin^2 theta - cos^2 theta)/(2 sin theta cos theta) = 3/5`

Answer 1

We have

`cos theta == 5/13`

In Δ ABC

`AC^2 = AB^2 + BC^2`

`=>(13)^2 = (AB)^2 + (5)^2`

`=> 169 = (AB)^2 + 25`

`=> (AB)^2 = 169 - 25`

=> AB = 12

`:. sin theta = 12/13 and tan theta = 12/5`

Now

`(sin^2 theta - cos^2 theta) xx  1/tan^2 theta = ((12/13)^2 - (5/13)^2)/(2 xx 12/13 xx 5/13) xx 1/(12/5)^2`

`= ((144 - 25)/169)/(120/169) xx 25/144`

`= 119/120 xx 25/144`

`= (119 xx 5)/(24 xx 144) = 595/3456`