Question

2 sec^2 theta - sec^4 theta - 2 cosec^2 theta + cosect^4 theta = cot^4 theta - tan^4 theta​

Answer 1

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.