Verify the identities please.
1.) TAN(X+π)-TAN(π-X)= 2 TAN X
2.) SIN(X+Y)+SIN(X-Y)= 2 SIN X COS Y
1. by the CAST rule
tan(π+Ø) = tanØ
tan(π-Ø) = -tanØ
LS = tan(x+π) - tan(π-x)
= tanx - (-tanx)
= 2tanx
= RS
2. again one of the standard identities used on the LS
LS = sinxcosy + cosxsiny + sinxcosy - cosxsiny
= 2sinxcosy
= RS
btw, 2sinxcosy = sin (2x) which would be a simplification of the RS
AWESOME! SUPERB! I had it right when you provided support. THANKS!
Sure! Let's verify these identities one by one.
1.) TAN(X+π) - TAN(π-X) = 2TAN(X)
To verify this identity, we need to simplify both sides of the equation and show that they are equal.
Starting with the left side of the equation:
TAN(X+π) - TAN(π-X)
Using the identity TAN(π + θ) = TAN(θ), we can rewrite the first term:
TAN(X+π) = TAN(X)
Next, using the identity TAN(π - θ) = -TAN(θ), we can rewrite the second term:
TAN(π-X) = -TAN(X)
Substituting these values back into the equation:
TAN(X) - (-TAN(X))
Since subtracting a negative is the same as adding a positive, we can simplify further:
TAN(X) + TAN(X)
And finally, we know that 2TAN(X) = TAN(X) + TAN(X), so:
TAN(X) + TAN(X) = 2TAN(X)
Therefore, we have verified that the identity is true.
2.) SIN(X+Y) + SIN(X-Y) = 2SIN(X)COS(Y)
To prove this identity, we'll simplify both sides of the equation and show that they are equal.
Starting with the left side of the equation:
SIN(X+Y) + SIN(X-Y)
Using the angle addition formula for sine, we can rewrite the first term:
SIN(X+Y) = SIN(X)COS(Y) + COS(X)SIN(Y)
Similarly, using the angle subtraction formula for sine, we can rewrite the second term:
SIN(X-Y) = SIN(X)COS(Y) - COS(X)SIN(Y)
Substituting these values back into the equation:
SIN(X)COS(Y) + COS(X)SIN(Y) + SIN(X)COS(Y) - COS(X)SIN(Y)
Rearranging the terms:
SIN(X)COS(Y) + SIN(X)COS(Y) + COS(X)SIN(Y) - COS(X)SIN(Y)
Since adding like terms gives us 2 of each term, we can simplify further:
2SIN(X)COS(Y) + (-2COS(X)SIN(Y))
Multiplying both terms by -1:
-2COS(X)SIN(Y) - 2COS(X)SIN(Y)
Since subtracting a negative is the same as adding a positive, we can simplify:
-2COS(X)SIN(Y) + 2COS(X)SIN(Y)
Finally, we observe that 2SIN(X)COS(Y) equals -2COS(X)SIN(Y), which verifies the given identity:
2SIN(X)COS(Y) = -2COS(X)SIN(Y)
Therefore, the second identity is also verified to be true.