 Hello and welcome to the session. In this session we discuss the following question which says, without using trigonometrical tables evaluate cos 58 degrees upon sin 32 degrees plus sin 22 degrees upon cos 68 degrees minus cos 38 degrees into the sec 52 degrees the whole upon tan 18 degrees into tan 35 degrees into tan 60 degrees into tan 72 degrees into tan 55 degrees. Let's move on to the solution now. We need to evaluate cos 58 degrees upon sin 32 degrees plus sin 22 degrees upon cos 68 degrees minus cos 38 degrees into cos 52 degrees the whole upon tan 18 degrees into tan 35 degrees into tan 60 degrees into tan 72 degrees into tan 55 degrees this is further equal to now cos 58 degrees can be written as cos 90 degrees minus 32 degrees upon sin 32 degrees plus sin 22 degrees can be written as sin 90 degrees minus 68 degrees upon cos 68 degrees minus cos 38 degrees into cos 90 degrees minus 38 degrees that is we have written cos 52 degrees as cos 90 degrees minus 38 degrees this whole upon tan 18 degrees into tan 35 degrees into now the value for tan 60 degrees is root 3 so we write root 3 here into tan 72 degrees can be written as tan 90 degrees minus 18 degrees into now tan 55 degrees can be written as tan 90 degrees minus 35 degrees here we have taken tan 60 degrees as root 3 so now this is further equal to cos 90 degrees minus 32 degrees is sin 32 degrees upon sin 32 degrees plus sin 90 degrees minus 68 degrees is cos 68 degrees upon cos 68 degrees minus cos 38 degrees into now cos 90 degrees minus 38 degrees can be written as sin 38 degrees this upon tan 18 degrees into tan 35 degrees into root 3 into now tan 90 degrees minus 18 degrees can be written as cos 18 degrees into tan 90 degrees minus 35 degrees is cos 35 degrees first as we know that cos of 90 degrees minus theta is equal to sin theta sin of 90 degrees minus theta is cos theta cos a cos 90 degrees minus theta is and tan of 90 degrees minus theta is cot theta. Now this is further equal to, this cancels with this so we have 1 plus, this cancels with this we have 1 minus. Now cot theta into sec theta is 1 so this is minus 1 upon tan 18 degrees into cot 18 degrees into root 3 into tan 35 degrees into cot 35 degrees. Now since we know that cot theta into sec theta is equal to 1 so here we get 1 and now this is further equal to 1 plus 1 minus 1 upon, now tan theta into cot theta is 1 into root 3 into 1 here we have used tan theta into cot theta is equal to 1 so we get this is equal to 2 minus 1 upon root 3 further we get 2 minus root 3 upon root 3 into root 3 that is this is equal to 2 minus root 3 upon 3 all you can say this is further equal to 6 minus root 3 upon 3 so the value of the given expression is 6 minus root 3 upon 3. 6 minus root 3 upon 3 is our final answer this completes the session where we have understood the solution of this question.