 Hello and welcome to the session. Let's discuss the following question. It says, evaluate without using trigonometric tables. So let us now move on to the solution. The given expression is cos square 25 degrees plus cos square 65 degrees upon cos square 65 degrees minus tan square 25 degrees plus tan 10 degrees got 36 degrees tan 80 degrees got 54 degrees upon tan square 30 degrees plus tan square 45 degrees. Now here we see that the sum of angles is 90 degrees. So we have cos square 25 degrees plus cos square 65 degrees can be written as cos square 90 minus 25 degrees. Upon again here the sum of the angles is 90 degrees. So cos square 65 degrees minus tan square 25 degrees can be written as tan square 90 minus 65 degrees because 25 can be written as 90 minus 65 plus tan 10 degrees. Here we see that 80 degrees gives us sum of angles here is 90 degrees. So we will convert one of these into cot and here it is cot 36 degrees and we will convert tan 80 degrees to cot tan degrees and for that what we do is tan 80 degrees can be written as tan 90 minus 10 degrees right into cot 54 degrees and cot 54 degrees can be written as cot 90 minus 36 degrees upon tan square 30 degrees plus tan square 45 degrees. Here we use the formula cos 90 minus theta is sin theta. So we have cos square 25 degrees plus sin square 25 degrees upon cos square 65 degrees minus tan square 90 minus 65 is cot square 65 degrees because tan 90 minus theta is cot theta plus tan 10 degrees cot 36 degrees tan 90 minus tan degrees is cot 10 degrees into cot 90 minus 36 degrees is tan 36 degrees upon tan square 30 degrees plus tan square 45 degrees. Here we have used the formula cos 90 minus theta is sin theta tan 90 minus theta is cot theta and we have used cot 90 minus theta is tan theta. This is very important to write the formula which we use now again. Now we know that cos square theta plus sin square theta is 1. So in the numerator we have 1 also consequence square theta minus cot square theta is 1. So we have 1 by 1. Now we have tan 10 degrees cot 10 degrees can be written as 1 upon tan 10 degrees into cot 36 degrees and tan 36 degrees can be written as 1 upon cot 36 degrees upon tan 30 is 1 upon root 3. So we have 1 upon root 3 square plus tan 45 degrees square is what 1 because tan 45 degrees is 1. And here we have used the formula cos square theta plus sin square theta is 1 consequence square theta minus cot square theta is 1 and tan theta is equal to 1 upon cot theta and vice-versa cot theta is 1 upon tan theta. So we have 1 plus 1 upon 1 by 3 plus 1. So this is equal to 1 plus 1 upon taking Lcm here we have 1 plus 3 upon 3. So this is equal to 1 plus 1 upon 4 upon 3 1 plus 3 by 4. So this is equal to 4 plus 3 by 4 taking Lcm. So this is equal to 7 by 4. Hence the value of the given expression is 7 by 4. So this completes the question and the session by for now take care have a good day.