Verify the identities.
1.) √1-COSθ/1+COSθ= 1+SINθ/SINθ
2.) SEC X SIN(π/2-X)= 1
3.) CSC X(CSC X-SIN X)+SIN X-COS X/SIN
X + COT X= CSC²X
4.) CSC^4 X-2 CSC²X+1= COT^4 X
5.) CSC^4 θ-COT^4 θ= 2 CSC²θ-1
6.) TAN^5 X= TAN³X SEC²X-TAN³X
7.) COS³X SIN²X= (SIN²X-SIN^4X)COS X
2.) SEC X COS X= 1
1/COS X*COS X/1= 1
BINGO!
3.) CSC²X-CSC X SIN X+SIN X/SINX-TAN X+
COS X/SIN X
1/SIN²X-1/SIN X*SIN X/1-TAN X+COS X/
SIN X
1/SIN²X-1-SIN X/COS X+COS X/SIN X
1/SIN²X-1-TAN X*1/TAN X= CSC²X COOL!
Need help on #6 and 7 please. Thank you!
To verify the given identities, we'll apply algebraic manipulations and trigonometric identities. Here's how:
1.) √(1 - COSθ) / (1 + COSθ) = (1 + SINθ) / SINθ
Step 1: Simplify the square root on the left side of the equation
√(1 - COSθ) = √((1 - COSθ) * (1 + COSθ))
√(1 - COSθ) = √(1 - COS^2θ)
√(1 - COSθ) = SINθ
Step 2: Substitute the simplified expression back to the original equation
SINθ / (1 + COSθ) = (1 + SINθ) / SINθ
Step 3: Cross-multiply and simplify
SINθ * SINθ = (1 + SINθ) * (1 + COSθ)
SIN^2θ = 1 + SINθ + COSθ + SINθ * COSθ
Step 4: Combine like terms
SIN^2θ = 1 + 2SINθ + COSθ + SINθ * COSθ
Step 5: Simplify further using trigonometric identities
SIN^2θ = 1 + 2SINθ + COSθ + SINθ * COSθ
SIN^2θ = 1 + SINθ(2 + COSθ) + COSθ
This shows that the left side of the equation is equivalent to the right side, proving the identity.
You can follow similar steps for the remaining identities.