Question

Prove it please
answer only if you know​

Answer 1

Part (c)

We'll use this identity

`\sin(x+y) = \sin(x)\cos(y) + \cos(x)\sin(y)\\\\`

to say

`\sin(A+45) = \sin(A)\cos(45) + \cos(A)\sin(45)\\\\\sin(A+45) = \sin(A)(√(2))/(2) + \cos(A)(√(2))/(2)\\\\\sin(A+45) = (√(2))/(2)(\sin(A)+\cos(A))\\\\`

Similarly,

`\sin(A-45) = \sin(A + (-45))\\\\\sin(A-45) = \sin(A)\cos(-45) + \cos(A)\sin(-45)\\\\\sin(A-45) = \sin(A)\cos(45) - \cos(A)\sin(45)\\\\\sin(A-45) = \sin(A)(√(2))/(2) - \cos(A)(√(2))/(2)\\\\\sin(A-45) = (√(2))/(2)(\sin(A)-\cos(A))\\\\`

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The key takeaways here are that

`\sin(A+45) = (√(2))/(2)(\sin(A)+\cos(A))\\\\\sin(A-45) = (√(2))/(2)(\sin(A)-\cos(A))\\\\`

Therefore,

`2\sin(A+45)*\sin(A-45) = 2*(√(2))/(2)(\sin(A)+\cos(A))*(√(2))/(2)(\sin(A)-\cos(A))\\\\2\sin(A+45)*\sin(A-45) = 2*\left((√(2))/(2)\right)^2\left(\sin^2(A)-\cos^2(A)\right)\\\\2\sin(A+45)*\sin(A-45) = 2*(2)/(4)\left(\sin^2(A)-\cos^2(A)\right)\\\\2\sin(A+45)*\sin(A-45) = \sin^2(A)-\cos^2(A)\\\\`

The identity is confirmed.

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Part (d)

`\sin(x+y) = \sin(x)\cos(y) + \cos(x)\sin(y)\\\\\sin(45+A) = \sin(45)\cos(A) + \cos(45)\sin(A)\\\\\sin(45+A) = (√(2))/(2)\cos(A) + (√(2))/(2)\sin(A)\\\\\sin(45+A) = (√(2))/(2)(\cos(A)+\sin(A))\\\\`

Similarly,

`\sin(45-A) = \sin(45 + (-A))\\\\\sin(45-A) = \sin(45)\cos(-A) + \cos(45)\sin(-A)\\\\\sin(45-A) = \sin(45)\cos(A) - \cos(45)\sin(A)\\\\\sin(45-A) = (√(2))/(2)\cos(A) - (√(2))/(2)\sin(A)\\\\\sin(45-A) = (√(2))/(2)(\cos(A)-\sin(A))\\\\`

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We'll square each equation

`\sin(45+A) = (√(2))/(2)(\cos(A)+\sin(A))\\\\\sin^2(45+A) = \left((√(2))/(2)(\cos(A)+\sin(A))\right)^2\\\\\sin^2(45+A) = (1)/(2)\left(\cos^2(A)+2\sin(A)\cos(A)+\sin^2(A)\right)\\\\\sin^2(45+A) = (1)/(2)\cos^2(A)+(1)/(2)*2\sin(A)\cos(A)+(1)/(2)\sin^2(A)\right)\\\\\sin^2(45+A) = (1)/(2)\cos^2(A)+\sin(A)\cos(A)+(1)/(2)\sin^2(A)\right)\\\\`

and

`\sin(45-A) = (√(2))/(2)(\cos(A)-\sin(A))\\\\\sin^2(45-A) = \left((√(2))/(2)(\cos(A)-\sin(A))\right)^2\\\\\sin^2(45-A) = (1)/(2)\left(\cos^2(A)-2\sin(A)\cos(A)+\sin^2(A)\right)\\\\\sin^2(45-A) = (1)/(2)\cos^2(A)-(1)/(2)*2\sin(A)\cos(A)+(1)/(2)\sin^2(A)\right)\\\\\sin^2(45-A) = (1)/(2)\cos^2(A)-\sin(A)\cos(A)+(1)/(2)\sin^2(A)\right)\\\\`

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Let's compare the results we got.

`\sin^2(45+A) = (1)/(2)\cos^2(A)+\sin(A)\cos(A)+(1)/(2)\sin^2(A)\right)\\\\\sin^2(45-A) = (1)/(2)\cos^2(A)-\sin(A)\cos(A)+(1)/(2)\sin^2(A)\right)\\\\`

Now if we add the terms straight down, we end up with
`\sin^2(45+A)+\sin^2(45-A)` on the left side

As for the right side, the sin(A)cos(A) terms cancel out since they add to 0.

Also note how
`(1)/(2)\cos^2(A)+(1)/(2)\cos^2(A) = \cos^2(A)` and similarly for the sin^2 terms as well.

The right hand side becomes
`\cos^2(A)+\sin^2(A)` but that's always equal to 1 (pythagorean trig identity)

This confirms that
`\sin^2(45+A)+\sin^2(45-A) = 1` is an identity