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

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

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

prove

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

False

Solution steps

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

Show that the two sides are not equal

For

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

plug in

x = 0

sin (\frac{π}{4} + 0) = \frac{2}{2} (Decimal : 0.70710 \dots)

sin (\frac{π}{4} + 0)

Simplify

= sin (\frac{π}{4})

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{2}{2}

sin (\frac{π}{4})

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= \frac{2}{2}

2 (cos (0) + sin (0)) = 2 (Decimal : 1.41421 \dots)

2 (cos (0) + sin (0))

Use the following trivial identity:

cos (0) = 1

cos (0)

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= 1

Use the following trivial identity:

sin (0) = 0

sin (0)

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 0

= 2 (1 + 0)

Simplify

= 2

The two sides are not equal

\Rightarrow False

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

arctan (x) = \frac{arcsin ( x )}{arccos ( x )}

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

1 + sin (4 a) + sin (2 a) = - 2 cos (a)

(csc^{2} (θ)) (tan^{2} (θ)) - 1 = tan^{2} (θ)

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