How is this solved using trig identities (sum/difference)?

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: