Verify the identities.
1.) SIN[(π/2)-X]/COS[(π/2)-X]=COT X
2.) SEC(-X)/CSC(-X)= -TAN X
3.) (1 + SIN Y)[1 + SIN(-Y)]= COS²Y
4.) 1 + CSC(-θ)/COS(-θ) + COT(-θ)= SEC θ
(Note: Just relax through verifying/solving these nice fun looking math problems! It's healthy for your brain!)
1.) 1/TAN[(π/2)-X]=COT X BINGO! SOLVED!
2.) SEC X/-CSC X
1/COS X ÷ -1/COS X
1/COS X * -SIN X/1
-TAN X YES BINGO! WOW!
(1+sin(y))(1+sin(-y)
(1+sin(y))(1-sin(y))
(1-sin^2(y))
cos^2(y)
I think the last one has a typo or needs some parentheses. If θ=pi/4,
1 + (-√2)/(1/√2) + (-1) = 1 - 2 - 1 = -2
but sec(pi/4) = √2
I think the last one should be
( 1 + csc(-Ø) / ( cos(-Ø) + cot(-Ø) ) = secØ
First of all , csc(-x) = -cscx and cot(-x) = -cotx , but cos(-x) = cosx
LS = (1 - cscx)/( cosx - cotx)
= (1 - 1/sinx) / (cosx - cosx/sinx)
= [ (sinx - 1)/sinx ] / [ (sinxcosx - cosx)/sinx ]
= (sinx - 1) / (sinxcosx - cosx)
= (sinx - 1) / (cosx(sinx - 1) )
= 1/cosx
= secx
= RS
Sure! Let's go through each identity and verify them:
1.) SIN[(π/2)-X]/COS[(π/2)-X] = COT X
To verify this identity, we need to simplify both sides of the equation and show that they are equal.
First, let's simplify the left side of the equation:
SIN[(π/2)-X] = COS(X) [using the sine angle difference identity]
COS[(π/2)-X] = SIN(X) [using the cosine angle difference identity]
So, the left side becomes:
COS(X)/SIN(X)
Now, let's simplify the right side of the equation:
COT X = COS(X)/SIN(X) [using the cotangent definition]
Both sides of the equation simplify to COS(X)/SIN(X). Hence, the left side is equal to the right side.
2.) SEC(-X)/CSC(-X) = -TAN X
To verify this identity, we need to simplify both sides of the equation and show that they are equal.
First, let's simplify the left side of the equation:
SEC(-X) = 1/COS(-X) = 1/COS(X) [since COS(-X) = COS(X), using the even property of cosine]
CSC(-X) = 1/SIN(-X) = -1/SIN(X) [since SIN(-X) = -SIN(X), using the odd property of sine]
So, the left side becomes:
(1/COS(X))/(-1/SIN(X)) = -SIN(X)/COS(X) = -TAN(X) [using the definition of tangent]
Hence, the left side is equal to the right side.
3.) (1 + SIN Y)(1 + SIN(-Y)) = COS²Y
To verify this identity, we need to simplify both sides of the equation and show that they are equal.
First, let's simplify the left side of the equation:
SIN(-Y) = -SIN(Y) [using the odd property of sine]
So, the left side becomes:
(1 + SIN(Y))(1 - SIN(Y)) = 1 - SIN²(Y) [using the difference of squares pattern]
Now, let's simplify the right side of the equation:
COS²Y = (1 - SIN²Y) [using the Pythagorean identity: SIN²Y + COS²Y = 1, rearranged to COS²Y = 1 - SIN²Y]
Hence, the left side is equal to the right side.
4.) 1 + CSC(-θ)/COS(-θ) + COT(-θ) = SECθ
To verify this identity, we need to simplify both sides of the equation and show that they are equal.
First, let's simplify the left side of the equation:
CSC(-θ) = 1/SIN(-θ) = -1/SIN(θ) [since SIN(-θ) = -SIN(θ), using the odd property of sine]
COS(-θ) = COS(θ) [since COS(-θ) = COS(θ), using the even property of cosine]
COT(-θ) = COS(-θ)/SIN(-θ) = COS(θ)/-SIN(θ) = -COT(θ) [using the properties of cosine and sine]
So, the left side becomes:
1 + (-1/SIN(θ))/COS(θ) - COT(θ) = 1 - (1/SIN(θ)) - COT(θ) = 1 - COT(θ) - COT(θ) = 1 - 2COT(θ)
Now, let's simplify the right side of the equation:
SECθ = 1/COSθ
Hence, the left side is not equal to the right side. The given equation is not an identity.
I hope this helps! Let me know if you have any further questions.