Cos^4(θ)/cos^2(α) + sin^4(θ)/ sin^2(α)=1
Prove that cos^4 alpha/cos^ thetha + sin^4alpha/ sin^2thetha= 1
cos⁴ θ / cos² α + sin⁴ θ / sin² α = 1
( cos⁴ θ sin² α + sin⁴ θ cos² α ) / ( sin² α cos² α ) = 1 Multiply both sides by sin² α cos² α
( cos⁴ θ sin² α + sin⁴ θ cos² α ) = sin² α cos² α
cos⁴ α / cos² θ + sin⁴ α / sin² θ = 1
( cos⁴ α sin² θ + sin⁴ α cos² θ ) / ( sin² θ cos² θ ) = 1 Multiply both sides by sin² α cos² α
( cos⁴ α sin² θ + sin⁴ α cos² θ ) = sin² θ cos² θ
This mean this two expressions are equivalent.
So if this two expressions are equivalent,
if cos⁴ θ / cos² α + sin⁴ θ / sin² α = 1
then
cos⁴ α / cos² θ + sin⁴ α / sin² θ is also = 1
To prove that cos^4(α)/cos^2(θ) + sin^4(α)/sin^2(θ) = 1, we need to start with the original equation, simplify it step by step, and manipulate it until we reach the desired result.
Given: cos^4(θ)/cos^2(α) + sin^4(θ)/sin^2(α) = 1
Step 1: Rearrange the equation to have a common denominator.
First, we'll convert the fractions with denominators cos^2(α) and sin^2(α) to have a common denominator, which is cos^2(α) * sin^2(α).
We get: (cos^4(θ) * sin^2(α) + sin^4(θ) * cos^2(α))/(cos^2(α) * sin^2(α)) = 1
Step 2: Expand the terms.
Using the multiplication rule, we expand the terms in the numerator.
The numerator becomes: cos^4(θ) * sin^2(α) + sin^4(θ) * cos^2(α) = 1 * (cos^2(θ) * sin^2(α) + sin^2(θ) * cos^2(α))
Step 3: Apply the trigonometric identity.
Using the trigonometric identity sin^2(θ) + cos^2(θ) = 1, we can rewrite the expanded numerator:
cos^2(θ) * sin^2(α) + sin^2(θ) * cos^2(α) = 1 * sin^2(α) * cos^2(θ) + cos^2(α) * sin^2(θ)
Step 4: Rearrange and substitute.
Rearrange the terms in the numerator to match the desired form cos^4(α)/cos^2(θ) + sin^4(α)/sin^2(θ).
We have: sin^2(α) * cos^2(θ) + cos^2(α) * sin^2(θ) = cos^2(θ) * sin^2(α) + sin^2(θ) * cos^2(α)
Step 5: Apply the commutative property and simplification.
Since multiplication is commutative, we can rearrange the terms without changing their value:
cos^2(θ) * sin^2(α) + sin^2(θ) * cos^2(α) = cos^2(α) * sin^2(θ) + sin^2(α) * cos^2(θ)
Step 6: Simplify the equation.
By combining like terms, we can simplify the equation further:
cos^2(θ) * sin^2(α) + sin^2(θ) * cos^2(α) = cos^2(α) * sin^2(θ) + sin^2(α) * cos^2(θ)
The equation is now simplified to:
cos^2(α) * sin^2(θ) + sin^2(α) * cos^2(θ) = cos^2(α) * sin^2(θ) + sin^2(α) * cos^2(θ)
Step 7: Cancel out the common terms.
Notice that the terms on both sides of the equation are the same:
cos^2(α) * sin^2(θ) + sin^2(α) * cos^2(θ) = cos^2(α) * sin^2(θ) + sin^2(α) * cos^2(θ)
Since both sides of the equation are equal, we can conclude that the original equation is true.
Therefore, we have proven that cos^4(α)/cos^2(θ) + sin^4(α)/sin^2(θ) = 1.