Is cos((5pi)/4-x)=-(sqrt(2))/2 (cos(x)+sin(x)) ?

The answer to whether cos((5pi)/4-x)=-(sqrt(2))/2 (cos(x)+sin(x)) is True

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prove cos((5pi)/4-x)=-(sqrt(2))/2 (cos(x)+sin(x))

prove

cos (\frac{5 π}{4} - x) = - \frac{2}{2} (cos (x) + sin (x))

True

Solution steps

cos (\frac{5 π}{4} - x) = - \frac{2}{2} (cos (x) + sin (x))

Manipulating left side

cos (\frac{5 π}{4} - x)

Rewrite using trig identities

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

Manipulating right side

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

Simplify

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

We showed that the two sides could take the same form

\Rightarrow True

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

5 cos^{2} (θ) + 3 sin^{2} (θ) = 3 + 2 cos^{2} (θ)

cos (x + \frac{π}{3}) + cos (x - \frac{π}{3}) = cos (x)

3 cos (x + y) + 3 cos (x - y) = 6 cos (x) cos (y)

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

Is cos((5pi)/4-x)=-(sqrt(2))/2 (cos(x)+sin(x)) ?
The answer to whether cos((5pi)/4-x)=-(sqrt(2))/2 (cos(x)+sin(x)) is True