Is cos(pi/4-x)+cos(pi/4+x)=sqrt(2)cos(x) ?

The answer to whether cos(pi/4-x)+cos(pi/4+x)=sqrt(2)cos(x) is True

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prove cos(pi/4-x)+cos(pi/4+x)=sqrt(2)cos(x)

prove

cos (\frac{π}{4} - x) + cos (\frac{π}{4} + x) = 2 cos (x)

True

Solution steps

cos (\frac{π}{4} - x) + cos (\frac{π}{4} + x) = 2 cos (x)

Manipulating left side

cos (\frac{π}{4} - x) + cos (\frac{π}{4} + x)

Rewrite using trig identities

= \frac{2 cos ( x ) + 2 sin ( x )}{2} + cos (\frac{π}{4} + x)

Rewrite using trig identities

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

Simplify

\frac{2 cos ( x ) + 2 sin ( x )}{2} + \frac{2 cos ( x ) - 2 sin ( x )}{2} : 2 cos (x)

= 2 cos (x)

We showed that the two sides could take the same form

\Rightarrow True

sin^{4} (x) - cos^{4} (x) = - cos (2 x)

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

\frac{cos ( x )}{tan ( x )} = csc (x) - sin (x)

sec^{2} (θ) csc^{2} (θ) = sec^{2} (θ) + csc^{2} (θ)

sin (θ) tan (θ) + cos (θ) = \frac{1}{cos ( θ )}

Is cos(pi/4-x)+cos(pi/4+x)=sqrt(2)cos(x) ?
The answer to whether cos(pi/4-x)+cos(pi/4+x)=sqrt(2)cos(x) is True