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Question

cos x cos pie/5-sin x sin pie/5= square root of 3/2

Answer 1

A reminder that

cos(a)cos(b) - sin(a)sin(b) = cos(a+b)

We apply this to the left hand side of the equation.

cos(x+pi/5) = sqrt(3)/2

Now, we take the cos(inverse) of both sides, while asking ourselves, where is cosine equal to sqrt(3)/2 on the unit circle? It occurs at pi/3 and 5pi/3

Therefore,

x + pi/5 = pi/3, 5pi/3
and
x = pi/3 + pi/5, 5pi/3 + pi/5

Note that this is only if you want the unit circle solutions. If you want the general solution,

x = (pi/3 + pi/5) + 2k(pi), for any integer k, and
= (5pi/3 + pi/5) + 2k(pi), for any integer k.

Answer 2

cos a cos b - sin a sin b = cos(a+b), so
cos(x + π/5) = (1/2)√3
x + π/5 = π/3 + 2kπ [since cos(π/3) = (1/2)√3 and the 2kπ since you didn't specifiy an interval for the solution.]
or
x + π/5 = -π/3 + 2kπ

x = 2π/15 + 2kπ, k = 0, 1, 2, ... or
x = -8π/15 + 2kπ, k = 0, 1, 2, ....