If `costheta= (4)/(5)` then find the value of `(sinthetacostheta+tan^(2)theta)`.

Explanation

Construct a right - angled `DeltaABC` in which `ltBAC=90^(@), AC=4k`
and BC=5k.
In `DeltaABC`,
From Pythagoras theorem
`AB^(2)+AC^(2)=BC^(2)`
`rArr " " AB^(2)=BC^(2)-AC^(2)=(5k)^(2)-(4k)^(2)`
` " " =25k^(2)-16k^(2)=9k^(2)`
`rArr " " AB=3k`
`:. " " sintheta=(AB)/(BC)=(3k)/(5k)=(3)/(5)`
`and" " tantheta=(AB)/(AC)=(3k)/(4k)=(3)/(4)`
Now , `sin thetacostheta+tan^(2)theta=(3)/(5)*(4)/(5)+((3)/(4))^(2)=(12)/(25)+(9)/(16)`
`" " (192+225)/(400)=(417)/(400)`