 Hello and welcome to the session the given question says without using trigonometric tables evaluate the following so this we have to evaluate let's start with the solution here we shall be using that sin 90 degrees minus theta is equal to cos theta and tan 90 degrees minus theta is equal to cot theta and here theta lies between 90 degree and 0 degree where 0 and 90 degrees are also included that is theta is less than or equal to 90 degree and theta is greater than or equal to 0 degrees and here 0 degree is less than theta s less than or equal to 90 degrees here we have to find the value of 2 times of sin 68 degrees divided by cos 22 degrees minus 2 times of cot 15 degree divided by 5 tan 75 degrees minus 3 times of tan 45 degrees into tan 20 degrees into tan 40 degrees into tan 50 degrees into tan 70 degrees whole divided by 5 now let us use these two identities here we have 2 times of sin 68 degrees which can be written as 2 times of sin 90 degrees minus 22 degrees divided by cos 22 degrees minus here we have 2 times of cot 15 degrees and in the denominator we have 2 5 times of tan 75 degrees which can be written as 90 degrees minus 15 degrees then we have minus 3 times of tan 45 degree is equal to 1 so into 1 into this can be written as tan 90 degrees minus 70 degrees into tan 40 degrees can be written as tan 90 degrees minus 50 degrees and then we have tan 50 degrees into tan 70 degrees whole divided by 5 now by using these two identities this can further be written as 22 degrees divided by cos 22 degrees minus 2 times of cot 15 degree divided by 5 times of cot 15 degree minus 3 tan 90 degrees minus 70 degrees is cot 70 degrees into similarly here we have cot 50 degree into tan 50 degree into tan 70 degree divided by 5 now here on cancelling we have 2 here we have 2 divided by 5 which we get on cancelling cos 15 degree by cot 15 degree and now since theta into tan theta is equal to 1 therefore cot 70 degree into tan 70 degree is equal to 1 and cot 50 degree into tan 50 degree is also 1 and the denominator we have 5 so we have 2 minus 2 divided by 5 minus 3 divided by 5 or this is further equal to taking 5 LCM we have 10 minus 2 minus 3 which is equal to 10 minus 5 divided by 5 or this is equal to 5 divided by 5 which is equal to 1 thus on evaluating we get the answer equal to 1 so this completes the solution by intake queue