Question

Use trig identities to change

\[ \frac{sec^3~\theta}{tan~\theta} \]
to
\[ sec~\theta~tan~\theta~+~csc~\theta \]

Answer 1

Hope it helps.

If you have any query, feel free to ask.

Answer 2

`RTP: (sec^(3)(x))/(tan(x)) = sec(x) \cdot tan(x) + cosec(x)`


`LHS = (sec^(3)(x))/(tan(x))`

`= (sec^(2)(x))/(tan(x)) \cdot sec(x)`


`sec^(2)(x) = tan^(2)(x) + 1`

`LHS = (tan^(2)(x) + 1)/(tan(x)) \cdot sec(x)`

`= (tan(x) + (1)/(tan(x))) \cdot sec(x)`

`= (tan(x) + cot(x)) \cdot sec(x)`

`= sec(x) \cdot tan(x) + cot(x) \cdot sec(x)`

`= sec(x) \cdot tan(x) + (1)/(cos(x)) \cdot (cos(x))/(sin(x))`

`= sec(x) \cdot tan(x) + (1)/(sin(x))`


`= sec(x) \cdot tan(x) + cosec(x)`

`= RHS`


`\therefore (sec^(3)(x))/(tan(x)) = sec(x) \cdot tan(x) + cosec(x)`