 Hello friends and how are you all today? The question says, choose the correct option and justify your choice. Now here we have four parts to the given question. We have choices with every part and we need to choose the correct option and justify our answer. Now here the first part which is given to us says that 2 tan 30 degree upon 1 plus tan square 30 degree should be equal to what? Should it be equal to sin 60 degree or cos 60 degree or tan 60 degree or sin 30 degrees. Now we will be doing our calculation in the rough part here. Now we know that the value of tan 30 degree is 1 by root 3 right. So 2 into 1 by root 3 upon 1 plus 1 by root 3 the whole square. So here after doing the calculation it will be 2 by root 3 getting divided by 1 plus 1 by 3. And which is equal to root 3 by 2 and we know the value of root 3 by 2 from the trigonometric ratios that we have learned in our table 8.1 is sin 60 degree. So here the answer to the first part will be sin 60 degree. Now in the same manner we will be proceeding with the rest of our parts. Here we have the second part 1 minus tan square 45 degree upon 1 plus tan square 45 degree. We know that the value of tan 45 degrees is 1. So we have 1 minus 1 the whole square upon 1 plus 1 the whole square which is further equal to 0 upon 2 which is further equal to 0. So the correct answer will be the D part and this is the required answer to the second part. Let us proceed with the third part. Here we have sin 2A is equal to 2 sin A is true when A is equal to now we know that when A is 0 degree we have sin 0 degree is equal to 2 into sin 0 degree and both of the sides will give us the answer as 0 isn't it? So here the first option is correct and this is the answer to the third part. Proceeding on with the last and final part. Here we have tan 30 is 1 by root 3 getting divided by 1 minus tan square 30 degree. It will be 1 by root 3 the whole square that will be equal to 1 by 3 isn't it? So we have 2 by root 3 divided by 2 by 3 which gives us the answer as root 3 and we know that the value of tan 60 degree is root 3. So here the answer to the fourth part will be tan 60 degree right? So let me write down the answers for you one again. So the correct options were for the first part it was A for the second part it was D for the third part it was A and for the fourth part it was C right? So this completes the session hope you understood it well. We have chosen the correct option and we have justified our answer side by side. So enjoy your life ahead.