Question

(cotθ-tanθ)/(sinθ+cosθ)=csc(θ)-sec(θ) verify

Answer 1

so you need to look at the numerator 1st

\[\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\]

\[\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\]

so then you are subtracting 2 fractions...

\[\cot(\theta) - \tan(\theta) = \frac{\cos(\theta)}{\sin(\theta)} - \frac{\sin(\theta)}{\cos(\theta)}\]

to subtract them you will now need a common denominator

\[\frac{\cos^2(\theta) - \sin^2(\theta)}{\sin(\theta)\cos(\theta)} \]

so you are now dealing with

\[\frac{\frac{\cos^2(\theta) - \sin^2(\theta)}{\sin(\theta)\cos(\theta)}}{\sin(\theta)+ \cos(\theta)}\]

which will result in

\[\frac{\cos^2(\theta) - \sin^2(\theta)}{\sin(\theta)\cos(\theta)(\sin(\theta) + \cos(\theta))}\]

my advice now would be to split the fractions identify common factors and simplify...

Answer 2

this may help.... someone may have an easier method

Answer 3

thats kind of how I started but than I got stuck and lost. Let me look at this again and see where can i go from here. thanks

Answer 4

i just continue what @campbell_st said