 Hello friends, so welcome to yet another problem-solving session on trigonometry now We have this question here. It's given that if tan square theta plus secant theta is 5 Then we have to find the value of cos theta okay, so our Relation has been given between tan theta and secant theta and we have to find out cos theta So how to go about it? We know that we have learned certain trigonometric identities. Yep So trigonometric identities will be used to solve this problem So basically if cos theta is to be found out we will resolve or you know reduce everything here like tan and secant in terms Of cos and let's see how it is done so Starting from the equation given so I can write tan square as Sin square theta by cos square theta plus 1 upon cos theta right and this is equal to 5 Why it secant is 1 upon cos and tan is sin upon cos Now going further so sin square theta using this identity sin square theta plus cos square theta is 1 you can express sin square as 1 minus cos square theta so hence 1 minus cos square theta divided by cos square theta plus The given 1 plus cos theta is 5. So this is step number 2 Going further I take LCM of the denominator going further. I take the LCM of the denominators I get cos square theta then Simplifying you'll get 1 minus cos theta cos square theta comes as it is and it is cos here and Denominator here is cos square. So cos will come over here, correct Now simplifying further 1 minus cos square theta plus cos theta is equal to this goes here So it becomes 5 cos square theta, correct now simplifying further So minus cos square theta here and 5 cos square theta So if you take all of them together on one side, you'll get 6 cos square theta This cos theta when goes on the other side it becomes minus cos theta and then and then Minus 1 equals to this. This is the final equation which you get if you see this is a quadratic equation Now if you see we have a quadratic equation in cos theta So let us assume cos theta to be x so hence the equation becomes 6x square minus x minus x minus 1 equals 0 Now What we have done here is if you know quadratic formula you can always find out The solution but we have used the splitting the middle term here And once you split the middle term the x minus x in the middle term can be expressed as minus 3x plus 2x Then you can factorize like this and finally you get a quadratic equation in this form factor in this form So hence we get x equals to half and x equals to minus 1 upon 3 now We'll also check whether we have done it correctly or not So if you see cos theta is half we find out tan theta which comes out to be root 3 and Then secant theta is equal to 1 upon cos that is 2 So hence if we deploy back those values in the given relation this one So we see that LHS comes out to be equal to RHS So when I do tan square theta plus secant theta it becomes 5 Similarly when you take cos theta is minus 3 then also you get tan square theta plus secant theta is 5 that indicates that both the solution that means cos theta is half and cos theta is minus 1 by 3 are correct