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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

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 (x) cot (x) sec (x) = 1

cot^{2} (a) + 1 = \frac{1}{sin ^{2} ( a )}

sin (θ) sec (θ) cot (θ) = 1

1 + sec^{2} (θ) sin^{2} (θ) = sec^{2} (θ)

3 sin^{2} (x) + 4 cos^{2} (x) = 3 + cos^{2} (x)

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