Is sin(θ)+cos(θ)=sqrt(2)sin(θ+45) ?

The answer to whether sin(θ)+cos(θ)=sqrt(2)sin(θ+45) is True

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prove sin(θ)+cos(θ)=sqrt(2)sin(θ+45)

prove

sin (θ) + cos (θ) = 2 sin (θ + 4 5^{\circ})

True

Solution steps

sin (θ) + cos (θ) = 2 sin (θ + 4 5^{\circ})

Manipulating right side

2 sin (θ + 4 5^{\circ})

Rewrite using trig identities

= 2 \frac{2 sin ( θ ) + 2 cos ( θ )}{2}

Simplify

2 \frac{2 sin ( θ ) + 2 cos ( θ )}{2} : sin (θ) + cos (θ)

= sin (θ) + cos (θ)

We showed that the two sides could take the same form

\Rightarrow True

\frac{1 - tan ( θ )}{1 + tan ( θ )} = \frac{cot ( θ ) - 1}{cot ( θ ) + 1}

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

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

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

sec (x) + csc (x) = csc (x) (tan (x) + 1)

Is sin(θ)+cos(θ)=sqrt(2)sin(θ+45) ?
The answer to whether sin(θ)+cos(θ)=sqrt(2)sin(θ+45) is True