#1 cos x+sec x=2
#2 (1+tan x)^2= 2 tan x+2
#3 cos^2x-sin^2 x+1=cos x
please show all work & thanks
cos x + 1/cos x = 2
cos^2 x -2 cos x + 1 = 0
(cos x - 1)(cos x - 1) = 0
cos x = 1
x = 0
Now you do some
Sure! Let's solve each of these equations step by step:
#1: To solve the equation cos(x) + sec(x) = 2:
Step 1: Rewrite sec(x) as 1/cos(x) to have only cosine terms:
cos(x) + 1/cos(x) = 2
Step 2: Multiply the entire equation by cos(x) to eliminate the denominators:
cos^2(x) + 1 = 2cos(x)
Step 3: Rearrange the equation and move all terms to one side:
cos^2(x) - 2cos(x) + 1 = 0
Step 4: Factor the equation:
(cos(x) - 1)(cos(x) - 1) = 0
Step 5: Set each factor equal to zero to solve for x:
cos(x) - 1 = 0 => cos(x) = 1
Step 6: The solution to cos(x) = 1 is x = 2πn, where n is an integer.
#2: To solve the equation (1 + tan(x))^2 = 2tan(x) + 2:
Step 1: Expand the square:
1 + 2tan(x) + tan^2(x) = 2tan(x) + 2
Step 2: Simplify the equation by canceling out like terms:
tan^2(x) - 2tan(x) + 1 = 0
Step 3: Factor the equation:
(tan(x) - 1)(tan(x) - 1) = 0
Step 4: Set each factor equal to zero to solve for x:
tan(x) - 1 = 0 => tan(x) = 1
Step 5: The solution to tan(x) = 1 is x = π/4 + πn, where n is an integer.
#3: To solve the equation cos^2(x) - sin^2(x) + 1 = cos(x):
Step 1: Rewrite cos^2(x) as (1 - sin^2(x)):
1 - sin^2(x) - sin^2(x) + 1 = cos(x)
Step 2: Combine like terms:
2 - 2sin^2(x) = cos(x)
Step 3: Rearrange the equation to one side:
2sin^2(x) + cos(x) - 2 = 0
To find the exact values of sin(x), cos(x), and tan(x), we need more information about x, such as a specific range or value.