Answer:
cos(2θ) = -1/2
i will assume you don't know about double angle identities and their alternate forms. an addendum is at the end of my work for the simpler method.
if
then we know that
because cosine is the reciprocal of secant (and vice-versa).
the information given to us of
also means that sine is negative because sine is the reciprocal of cosecant, and reciprocal does not change the sign.
so we know cosine is positive and sine is negative. therefore this must be an angle in quadrant iv.
we know that cos θ = 1/2. and cos θ = adjacent/hypotenuse. we can figure out sin θ from this information.
if we draw an angle in quadrant 4, then we can form a reference right-angle triangle with an angle theta. we have the side adjacent to it be 1 and the hypotenuse be 2, then using the pythagorean theorem to figure out opposite side would result in
but we are given the sine is negative, therefore
(see diagram; it does contain alternate text but still applicable.)
cosine addition identity:
for , we can rewrite as and therefore it becomes
since we know
then substituting in
so cos(2θ) = -1/2
there is a simpler way to do this that does not involve finding sine of theta. we concluded that
consider the pythagorean identity ; subtracting both sides by cosine squared results in
substituting into the above:
substituting info of cosine:
