Evaluate or simplify expressions

Question

I had three problems I can’t figure out at all.

1. Sec^-1(Csc theta)
2. Cos^-1x+ Cos^-1-X
3. Tan^-1(Cot theta)

Answer 1

1. sec^-1(cscθ) = sec^-1(sec(π/2 - θ)) = π/2; 0 < θ < π
2. cos^-1x + cos^-1(- x) = cos^-1x + π - cos^-1x = π
3. tan^-1(cotθ) = tan^-1(tan(π/2 - θ)) =π/2 - θ; 0 < θ < π

Answer 2

wouldn't this work -- note added parts in bold, and k=0 or +/-1,+/-2, ...

sec^-1[cscθ]=sec^-1[sec(π/2-θ]

=π/2-θ+2πk=(π/2)*(4k+1)-θ

or sec^-1[cscθ] =(π/2+θ)+2πk=(π/2)*(4k+1)+θ

for principal value of sec^-1, k must be chosen so 0<sec^-1[cscθ]<π

same with tan^-1cot(θ) sort of

tan^-1cot(θ)=π/2-θ+πk=(π/2)*(2k+1)-θ

for principal value of tan^-1cot(x), k must be chosen so -π/2<tan^-1cot(x)<π/2

for cos^-1x+cos^-1(-x)

cos^-1x equals some angle, call it β and actually it is +/- β

cos^-1(-x) equals same +/- β but π radians away from original +/- β

so cos^-1x+cos^-1(-x) could equal π or equal either π+/- 2*β

if principal value is meant, then cos^-1x>0 and cos^-1(-x)>0

and cos^-1x+cos^-1x=π