tan35cot55 - sec35csc55 + cot210

Simplify the ratios and the recipricols:

tan is `sin/cos` and cot is `1/tan` which means cot is the recipricol of tan `therefore 1/tan = cos/sin`

also sec is `1/cos` and csc is `1/sin`

cot is `1/tan` as previously mentioned and the angle is greaater than 180 degrees therefore:

`(cos 210)/(sin210) = (cos 180 +30)/sin(180 +30)`

`therefore cot 210 = cos30/sin30`

Now to put all this together:

tan35cot55-sec35csc55+cot210

= `sin35/cos35 times cos55/sin55 - 1/cos35 times 1/sin55 + cos30/sin30`

we also know that sin and cos are co-ratios of each other.For example, cos 35 = sin 55 (the two add up to 90 degrees). So we simplify further:

= `sin35/cos35 times sin35/cos35 - 1/cos35 times 1/sin55 + cos 30/sin30`  

note how we used the co-ratios (sin 35= cos 55)

we can do the same thing with the `1/cos35 times 1/sin 55` and we can multiply our first term:

= `(sin^2 35)/(cos ^2 35)- 1/cos35 times 1/cos35 + cos 30/sin30`

We can also simplify cos 30 and sin 30 as they are special angles. Sin 30 = `1/2` and cos 30 = `sqrt3/2` so  for 

`cos 30/sin30` we have `sqrt3/2 divide 1/2` which = `sqrt3/2 times 2/1 = sqrt3`

`therefore = (sin^2 35)/(cos^2 35) - 1/(cos^2 35) + sqrt3`

We know from identities that `sin^2 35 + cos ^2 35 = 1` so we can substitute for the 1 in our expression:

`therefore = (sin ^2 35)/(cos^2 35) - ((sin^2 35 + cos^2 35))/(cos^2 35)+sqrt3`

Use the common denominator which is `cos^2 35` and remove brackets:

`therefore = (sin^2 35 - sin^2 35 - cos^2 35 + sqrt3(cos ^2 35))/ (cos^2 35)`

Note how the symbols changed when we removed the brackets. Now simplify:

`therefore = (-cos^2 35 +sqrt3cos^2 35)/ (cos^2 35)`  now factorize:

`therefore = (cos^2 35( -1 +sqrt3))/ (cos^2 35)`

`therefore = -1 + sqrt3`

`therefore = ` 0.73 (rounded off to two decimal places)