Suppose that sec alpha = 5/4 and tan beta = -1, where 3pi/2<alpha<beta<2pi. Find cos(alpha + beta).

Question

Simplify the expression

Answer 1

the angle is in quadrant IV

where sines are negative, cosines positive

tanB -1 means sinB=-1/sqr2 and cosB = 1/sqr2

secA = 5/4 means cosA =4/5 and sinA = -3/5

cos(A+B) = cosAcosB+ sinAsinB = (4/5)(1/sqr2)+(-3/5)(-1/sqr2) = 7/5sqr2

Answer 2

sec α = 5/4 ⇒ cos α = 4/5 ⇒ sin α = 3/5

tan β = -1 ⇒ sin -√2/2 ⇒ cos β = √2/2

Now substitute values in the angle-sum formula.