What is the general solution for (cos(x)+cos(3x))/(cos^2(x)-sin^2(x))=0 ?

The general solution for (cos(x)+cos(3x))/(cos^2(x)-sin^2(x))=0 is x= pi/2+2pin,x=(3pi)/2+2pin

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(cos(x)+cos(3x))/(cos^2(x)-sin^2(x))=0

\frac{cos ( x ) + cos ( 3 x )}{cos ^{2} ( x ) - sin ^{2} ( x )} = 0

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Solution steps

\frac{cos ( x ) + cos ( 3 x )}{cos ^{2} ( x ) - sin ^{2} ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

cos (x) + cos (3 x) = 0

Rewrite using trig identities

2 cos (2 x) cos (x) = 0

Solving each part separately

cos (2 x) = 0 or cos (x) = 0

cos (2 x) = 0 : x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Combine all the solutions

x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn, x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Since the equation is undefined for:

\frac{π}{4} + πn, \frac{3 π}{4} + πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

sin (x) = 2 cos^{2} (x) - 1

tan (t) = \frac{- 5}{12}

\frac{( 1 - ( cos ( x ) ) ^{2} )}{( 1 + sin ( x ) )} = sin (x)

x, arctan (x) = 0

5 sin (x) = - 3

What is the general solution for (cos(x)+cos(3x))/(cos^2(x)-sin^2(x))=0 ?
The general solution for (cos(x)+cos(3x))/(cos^2(x)-sin^2(x))=0 is x= pi/2+2pin,x=(3pi)/2+2pin