 Hello and welcome to the session. In this session we discuss the following question which says without using trigonometric tables evaluate the following cos x square 90 degrees minus theta minus tan square theta upon 4 into sin square 52 degrees plus sin square 38 degrees plus 2 tan square 45 degrees into cos x square 52 degrees into cos square 38 degrees upon cos x square 70 degrees minus tan square 20 degrees. Let's proceed with the solution now. We need to evaluate cos x square 90 degrees minus theta minus tan square theta upon 4 into sin square 52 degrees plus sin square 38 degrees plus 2 tan square 45 degrees into cos x square 52 degrees into cos square 38 degrees this whole upon cos x square 70 degrees minus tan square 20 degrees. First let's consider this expression that is cos x square 90 degrees minus theta minus tan square theta upon 4 into sin square 52 degrees plus sin square 38 degrees. Let us first evaluate this expression. Now this is equal to in the numerator we have sin square theta minus tan square theta upon 4 into sin square 90 degrees minus 38 degrees plus sin square 38 degrees. Since we know that cos x 90 degrees minus theta is equal to sin theta so cos x square 90 degrees minus theta can be written as sin square theta also we have written sin 52 degrees as sin 90 degrees minus 38 degrees and sin square 52 degrees can be written as sin square 90 degrees minus 38 degrees. Further this is equal to 1 upon 4 into cos square 38 degrees plus sin square 38 degrees. Since we know that sin square theta is equal to 1 plus tan square theta so from where we get that's x square theta minus tan square theta is equal to 1 so in the numerator we have got 1 then again we have that sin 90 degrees minus theta is equal to cos theta so sin square 90 degrees minus theta is equal to cos square theta so sin square 90 degrees minus 38 degrees is equal to cos square 38 degrees. So this is equal to 1 upon 4 since we know that sin square theta plus cos square theta is equal to 1 so sin square 38 degrees plus cos square 38 degrees is equal to 1. So therefore we have got cos x square 90 degrees minus theta minus tan square theta upon 4 into sin square 52 degrees plus sin square 38 degrees is equal to 1 upon 4. Now consider the other expression which is 2 tan square 45 degrees into cos x square 52 degrees into cos square 38 degrees this whole upon cos x square 70 degrees minus tan square 20 degrees. Now this can be further written as 2 into 1 since we know tan 45 degrees is 1 so tan square 45 degrees is 1 into now we can write cos x square 52 degrees as cos x square 90 degrees minus 38 degrees into cos square 38 degrees this whole upon cos x square 70 degrees can be written as cos x square 90 degrees minus 20 degrees minus tan square 20 degrees this is further equal to 2 into sex square 38 degrees into cos square 38 degrees upon sex square 20 degrees minus tan square 20 degrees since we know that cos x 90 degrees minus theta is equal to sex theta so cos x square 90 degrees minus 38 degrees is written as sex square 38 degrees and also cos x square 90 degrees minus 20 degrees can be written as sex square 20 degrees so now this is further equal to 2 into 1 upon cos square 38 degrees into cos square 38 degrees this whole upon 1 since we know that sex square theta is equal to 1 plus tan square theta so sex square theta minus tan square theta is 1 and also we know that sex theta is equal to 1 upon cos theta so we have written sex square 38 degrees as 1 upon cos square 38 degrees so we are left with 2 upon 1 that is equal to 2 thus we get 2 tan square 45 degrees into cos x square 52 degrees into cos square 38 degrees this whole upon cos x square 70 degrees minus tan square 20 degrees is equal to 2 so this is the given expression this would be equal to value of this expression which was 1 upon 4 plus the value of this expression which is 2 so 1 upon 4 plus 2 this is equal to 9 upon 4 so 9 upon 4 is our final answer this completes the session hope you have understood the solution of this question