Is sin(x)+cos(x)=sqrt(2) ?

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

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prove sin(x)+cos(x)=sqrt(2)

prove

sin (x) + cos (x) = 2

False

Solution steps

sin (x) + cos (x) = 2

Show that the two sides are not equal

For

sin (x) + cos (x) = 2

plug in

x = 0

sin (0) + cos (0) = 1

sin (0) + cos (0)

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

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

= 0 + 1

Simplify

= 1

The two sides are not equal

\Rightarrow False

sin^{2} (x) + \frac{1}{sec ^{2} ( x )} = sin (x) csc (x)

sin (x + x) = 2 sin (x) cos (x)

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

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

csc (2 x) - cot (2 x) = tan (2 x)