Question

Cos(5п/2-x) -2cos2x=1

Answer 1

Cos(5п/2-x) -2cos2x=1
sinx-2cos2x-1=0
sinx-2+4sin²x-1=0
sinx=a
4a²+a-3=0
D=1+48=49
a1=(-1-7)/8=-1⇒sinx=-1⇒x=-π/2+2πk,k∈z
a2=(-1+7)/8=3/4⇒sinx=3/4⇒x=(-1)^karcsin3/4+πk,k∈z

Answer 2

To solve the equation cos(5π/2 - x) - 2cos(2x) = 1, we can simplify and manipulate the equation using trigonometric identities. Let's break it down step by step:

1. Rewrite the cos(5π/2 - x) term using the cosine difference formula: cos(5π/2 - x) = cos(5π/2)cos(x) + sin(5π/2)sin(x) Since cos(5π/2) = 0 and sin(5π/2) = -1, we have: cos(5π/2 - x) = 0*cos(x) + (-1)*sin(x) = -sin(x)

2. Substitute the simplified expression back into the equation: -sin(x) - 2cos(2x) = 1

3. Rearrange the terms: -sin(x) = 1 + 2cos(2x)

4. Apply the double angle formula for cosine: cos(2x) = 2cos^2(x) - 1 Substitute this back into the equation: -sin(x) = 1 + 2(2cos^2(x) - 1)

5. Simplify: -sin(x) = 1 + 4cos^2(x) - 2 -sin(x) = 4cos^2(x) - 1

6. Rearrange the terms: 4cos^2(x) + sin(x) - 1 = 0

Now, you can solve this quadratic equation for cos(x) using numerical methods or trigonometric techniques. Unfortunately, it's not possible to find a simple exact solution for this equation using algebraic methods alone.