Question

Cos^2(pi-x)-sin(pi/2-x)=0

Answer 1

Cos^2(pi-x)-sin(pi/2-x)=0
По формулам приведения получаем.
cos^2 x - cos x =0
cosx(cosx-1)=0
1). cosx=0
x=pi/2+pin, n принадлежит Z.
2). cosx-1=0
cosx=1
x= 2pik, k принадлежит Z.

Answer 2

The equation you provided is: cos^2(π - x) - sin(π/2 - x) = 0

To solve this equation, let's break it down step by step:

Step 1: Simplify the equation using trigonometric identities.

First, let's simplify the term cos^2(π - x) using the identity: cos^2(a) = 1 - sin^2(a).

So, cos^2(π - x) = 1 - sin^2(π - x).

Next, let's simplify the term sin(π/2 - x) using the identity: sin(a - b) = sin(a) * cos(b) - cos(a) * sin(b).

So, sin(π/2 - x) = sin(π/2) * cos(x) - cos(π/2) * sin(x).

Since sin(π/2) = 1 and cos(π/2) = 0, we can simplify further:

sin(π/2 - x) = 1 * cos(x) - 0 * sin(x) = cos(x).

Now, our equation becomes:

1 - sin^2(π - x) - cos(x) = 0.

Step 2: Apply the Pythagorean identity.

The Pythagorean identity states that sin^2(a) + cos^2(a) = 1.

Using this identity, we can rewrite the equation as:

1 - (1 - cos^2(π - x)) - cos(x) = 0.

Simplifying further, we get:

1 - 1 + cos^2(π - x) - cos(x) = 0.

Step 3: Combine like terms.

cos^2(π - x) - cos(x) = 0.

Step 4: Factor out common terms.

We can factor out cos(x) from both terms:

cos(x) * (cos(π - x) - 1) = 0.

Step 5: Solve for x.

To find the values of x that satisfy the equation, we set each factor equal to zero and solve for x.

First, let's consider the factor cos(x) = 0. This means that x is an angle where the cosine function equals zero. The solutions for this factor can be found by setting x to π/2 + kπ, where k is an integer.

Second, let's consider the factor cos(π - x) - 1 = 0. Solving this equation, we get:

cos(π - x) = 1.

Using the identity cos(a) = 1 when a = 0, we have:

π - x = 0.

Solving for x, we get:

x = π.

Therefore, the solutions to the equation cos^2(π - x) - sin(π/2 - x) = 0 are:

x = π/2 + kπ (where k is an integer) and x = π.