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

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

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

prove

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

True

Solution steps

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

Manipulating left side

sin (\frac{π}{4}) + cos (\frac{π}{4})

Simplify

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

cos (\frac{π}{4}) + sin (\frac{π}{4})

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

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}

= \frac{2}{2} + sin (\frac{π}{4})

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{2}{2}

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}

Simplify

\frac{2}{2} + \frac{2}{2}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 + 2}{2}

Add similar elements:

2 + 2 = 22

= \frac{2 2}{2}

Divide the numbers:

\frac{2}{2} = 1

= 2

= 2

= 2

We showed that the two sides could take the same form

\Rightarrow True

cos (- θ) csc (- θ) tan (- θ) = 1

cos (θ) (tan (θ) - tan (\frac{θ}{2})) = tan (\frac{θ}{2})

sec (x) - 2 tan (x) = 0

cos (3 θ) = cos (θ) (cos^{2} (θ) - 3 sin^{2} (θ))

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

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