Question

solve the problem very easy steps

Answer 1

You're supposed to bring sec(theta) and tan(theta) in the denominators, so think of what can be done to do so. 

You'll come to a conclusion that if we'll divide numerator and denominator by cos(theta) you'll get something relevant..., here it goes, 

take LHS

multiply and divide the num. and den. with cos(theta)...

(sinФ/ cosФ) - (cosФ/cosФ)  + (1/cosФ)           tanФ - 1 + secФ
-------------------------------------------------------- =  -------------------------------     -----{1}
(sinФ/cosФ) + (cosФ/cosФ) - (1/cosФ)             tanФ + 1 - secФ

we know that, sec^2(Ф) = 1 + tan^2(Ф)
                       thus, sec^2(Ф) - tan^2(Ф) = 1     ----{2}
Now, put {2} in {1}

[Note: Remember, in such questions do not substitute the value of 1 in the denominator. I'll be a better choice;) ]

Now,

tanФ + secФ - (sec^2(Ф) - tan^2(Ф))                (tanФ + secФ)[ 1 - secФ + tanФ]
-----------------------------------------------------   =        --------------------------------------------
                  tanФ + 1 - secФ                                    [1 - secФ + tanФ]

     = tanФ + secФ

Multiply and divide this by tanФ - secФ.
[Note: in such questions, if you land with the answer of this type, trying this method shall prove to be better. ]

On further solving you'll land-up with RHS;)