How do I prove [math]\sec^4\theta – \cos^4\theta = 1 – 2\cos^2\theta[/math]?
Let’s try messing around with the equation.
1/cos4 (θ) −cos4(θ) = 1−2cos2(θ)
Multiply both sides by cos4(θ)
1−cos8(θ)=cos4(θ)−2cos6(θ)
Let u=cos2(θ)
1−u4=u2−2u3
u4−2u3+u2−1=0(1)
Since this is a degree 4 equation, it can have at most 4 solutions. That means that cos2(θ) can have at most 4 values that satisfy (1) . Since only 4 values can be satisfied at max, and |R|≫4 , your original equation is not true for the vast majority of θ values.
Answer:
How do I prove [math]\sec^4\theta – \cos^4\theta = 1 – 2\cos^2\theta[/math]?
Let’s try messing around with the equation.
1/cos4 (θ) −cos4(θ) = 1−2cos2(θ)
Multiply both sides by cos4(θ)
1−cos8(θ)=cos4(θ)−2cos6(θ)
Let u=cos2(θ)
1−u4=u2−2u3
u4−2u3+u2−1=0(1)
Since this is a degree 4 equation, it can have at most 4 solutions. That means that cos2(θ) can have at most 4 values that satisfy (1) . Since only 4 values can be satisfied at max, and |R|≫4 , your original equation is not true for the vast majority of θ values.