Is (2sin(x)+cos(x))^2+(2cos(x)-sin(x))^2=5 ?

The answer to whether (2sin(x)+cos(x))^2+(2cos(x)-sin(x))^2=5 is True

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prove (2sin(x)+cos(x))^2+(2cos(x)-sin(x))^2=5

prove

(2 sin (x) + cos (x))^{2} + (2 cos (x) - sin (x))^{2} = 5

True

Solution steps

(2 sin (x) + cos (x))^{2} + (2 cos (x) - sin (x))^{2} = 5

Manipulating left side

(2 sin (x) + cos (x))^{2} + (2 cos (x) - sin (x))^{2}

Factor

(- sin (x) + 2 cos (x))^{2} + (cos (x) + 2 sin (x))^{2} : 5 (sin^{2} (x) + cos^{2} (x))

= (cos^{2} (x) + sin^{2} (x)) \cdot 5

Rewrite using trig identities

= 5

We showed that the two sides could take the same form

\Rightarrow True

cos (a) (sec (a) - cos (a)) = sin^{2} (a)

tan (x) cos (x) csc (x) = 1

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

\frac{csc ^{4} ( x ) - 1}{1 + csc ^{2} ( x )} = cot^{2} (x)

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

Is (2sin(x)+cos(x))^2+(2cos(x)-sin(x))^2=5 ?
The answer to whether (2sin(x)+cos(x))^2+(2cos(x)-sin(x))^2=5 is True