Use the trigonometric subtraction formula for sine to verify this identity cos(( (\pi)/(2) ) - x) = sinx ​

Question

Use the trigonometric subtraction formula for sine to verify this identity


`cos(( (\pi)/(2) ) - x) = sinx`
​

Answer 1

The cosine difference formula yields

`\cos(a-b)=\cos(a)\cos(b)+\sin(a)\sin(b)`

In your case,

`a=(\pi)/(2),\quad b=x`

so the formula translates to

`\cos\left((\pi)/(2)-x\right)=\cos\left((\pi)/(2)\right)\cos(x)+\sin\left((\pi)/(2)\right)\sin(x)`

Since

`\cos\left((\pi)/(2)\right)=0,\quad \sin\left((\pi)/(2)\right)=1`

the expression above becomes

`0\cdot\cos(x)+1\cdot\sin(x)=\sin(x)`