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csc(θ)>0\land sec(θ)<0

Solution

csc (θ) > 0 and sec (θ) < 0

False for all θ \in R

Solution steps

csc (θ) > 0 and sec (θ) < 0

csc (θ) > 0 :

False for all

θ \in R

csc (θ) > 0

Express with sin, cos

csc (θ) > 0

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

\frac{1}{sin ( θ )} > 0

\frac{1}{sin ( θ )} > 0

\frac{1}{a} > 0

then

a > 0

sin (θ) > 0

No Solution for θ \in R

sec (θ) < 0 :

False for all

θ \in R

sec (θ) < 0

Express with sin, cos

sec (θ) < 0

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

\frac{1}{cos ( θ )} < 0

\frac{1}{cos ( θ )} < 0

\frac{1}{a} < 0

then

a < 0

cos (θ) < 0

No Solution for θ \in R

Combine the intervals

False for all θ \in R and False for all θ \in R

Merge Overlapping Intervals

False for all θ \in R and False for all θ \in R

The intersection of two intervals is the set of numbers which are in both intervals

False for all

θ \in R

and

False for all

θ \in R

False for all θ \in R

False for all θ \in R

tan (θ) < 0 and cos (θ) > 0

sec (θ) = \frac{4}{3} and cot (θ) < 0

0 < cos (x) < sin (x)

\frac{1}{2} < sin (θ) < \frac{3}{2}

\frac{x ^{2}}{2} - sin^{2} (x) i n^{0} < x < 2 π