Question

Plz prove lhs=rhs thanks

Answer 1

Identities are essentially equations that are true for all input values in the domain. A simple identity you'd find in a previous algebra class would be something like x+x = 2x. No matter what value you replace for x, it will be a true equation. The same applies here but on a slightly more complicated level. 

To prove identities are true, the goal is to get one side to transform into the other. Whatever side you pick to transform will have the other side stay the same no matter what happens. I'm going to pick the left side to transform while the right hand side stays the same. 

I'll be taking advantage of the pythagorean identity sec^2(x) - tan^2(x) = 1 which is a rearrangement of sec^2(x) = tan^2(x) + 1 (derived from sin^2(x) + cos^2(x) = 1)

It may not be obvious, but if we multiply top and bottom of the fraction by sec(theta)-tan(theta), then the denominator will turn into sec^2(theta) - tan^2(theta) which simplifies to 1 using the previously mentioned identity. I used the difference of squares rule. 

The numerator will use the rule (a-b)^2 = a^2-2*a*b+b^2 to expand out.

Have a look at the attached image to see the full step-by-step breakdown. 

Notes:
The red portion in step 2 is done to highlight the change (so it doesn't look so cluttered)
The blue portion further down is to highlight how sec^2 turns into 1+tan^2