 Hello, welcome to another session on problem solving on trigonometry. So the question here says if secant theta is equal to x plus 1 by 4x prove that secant theta plus tan theta is equal to 2x or 1 by 2x. So this is what you have to do. So it's given that secant theta is equal to x plus 1 by 4x and you have to prove that secant theta plus tan theta is equal to 2x or 1 upon 2x. Now in such questions, how do we really approach them? So if you see the thing to be proved here is secant plus tan. So the moment there is a combination of secant and tan, one thing which strikes our mind is the identity, trigonometric identity that is secant square theta minus tan square theta is equal to 1. So this is the first thing which comes to our mind. Now we have to use this particular trigonometric identity to solve this one. It's not necessary that only this identity has to be used to solve the question. But since secant and tan is involved, so first thing which comes in our mind is secant square minus tan square theta is equal to 1. And second, other thing which is given, which is also giving us a direction is the presence of x the presence of x and then another term which is 1 upon 4x. Now you would have done in previous grades where if you square x plus 1 by x whole, if you do the square of let us say terms like x plus 1 by x whole square, then what happens is it is very easy to convert this back into x minus 1 by x whole square like that. Yeah, why? Because squaring you will get x square plus 1 upon x square plus 2. Now you can do mathematical manipulation to make it minus 2. So it becomes x square plus 1 by x square minus 2 plus 4. It can be written like that, isn't it? Which is nothing but x minus 1 by x whole square minus 2 square. So this item is same as this. This is very important. I think so whenever we see terms like x and 1 by 4x and or x plus 1 by x and terms like that, then this is again, which comes to our mind. Now using these two only will be solving this question. So let us see how. So secant square theta minus tan square theta is equal to 1, right? So tan square theta will be nothing but secant square theta minus 1. And now we deploy the value of secant theta in this expression, which is given here. So we deploy this value of secant theta, which is given here into this equation. So you will get x square plus 1 by 16 x square plus 2 times x into 1 by 4x minus 1. And this is nothing but a plus b whole square is equal to a square plus b square plus twice a b. This is used in this step. OK. Now if you simplify it further, you will get this x gets cancelled. So you will get 2 by 1 by 4. So 2 into 1 by 4 is 1 by 2 and then minus 1. So hence this will be after simplification. Now this is a clear cut indication that is something which was plus half has become 1 minus half. That means again I can reduce this to x minus 1 by 4x whole square form. How? So if you see the logical next step is what you can do. You can write 1 by 2 as 2 times x into 1 by 4x. This is purposefully done. Why? Because this will lead to the step where now we started from x plus 1 by 4x whole square. Now we are getting x minus 1 by 4x square. This is very important. Now where did we start from? Tan square theta. Is it it? So hence this will be tan theta. If you take, you know, solve this equation, which equation? Tan square theta is equal to this. So tan theta will be either plus x minus 1 by 4x or minus x minus 4x. Why? Because if you have an expression like y square is equal to c square, then y is equal to either plus c or minus c. Two equations, two solutions are there. Now, so there are two possibilities of tan theta now. So we have, we had to find out secant theta plus tan theta, this. So secant theta already was known, which was given in this. So you have to now add tan theta, which you just found out here. So secant theta plus tan theta will be x plus 1 by 4x plus x minus 1 by 4x. Or if tan theta is minus x minus 4x, then this is a solution. So if you add the first, first set of terms, then you will get 2x. And in this case, you will get 1 by 2x. And this is what we had to show. So either tan theta is 2x or 1 upon 2x. So we got it. So what is the learning? Learning is keep the relevant identity in your mind. So secant square theta minus tan square theta was the starting point. Why? Because the question was demanding some relation between secant and tan. So that was one thing. And secondly, this x plus 1 by 4x form was directing us to square it and get x minus 1 by 4x, a term containing x minus 1 by 4x. And this actually helped us to solve this problem. Keep this in mind. Thank you.