Prove cos^2t+4cost+4/cost+2=2sect+1/sect
To prove the equation cos^2(t) + 4cos(t) + 4/cos(t) + 2 = 2sec(t) + 1/sec(t), we can simplify both sides of the equation separately and verify if they are equal.
Let's start with the left-hand side (LHS):
LHS: cos^2(t) + 4cos(t) + 4/cos(t) + 2
Step 1: Combine the cosine terms:
cos^2(t) + 4cos(t) = (cos(t))^2 + 2(2)cos(t) = (cos(t) + 2)^2
Step 2: Rewrite 4/cos(t) as 4sec(t):
LHS = (cos(t) + 2)^2 + 4sec(t) + 2
Now let's simplify the right-hand side (RHS):
RHS: 2sec(t) + 1/sec(t)
Step 1: Combine sec(t) terms by finding the common denominator, which is sec(t):
RHS = (2sec^2(t) + 1) / sec(t)
Step 2: Use the identity sec^2(t) = 1 + tan^2(t):
RHS = (2(1 + tan^2(t)) + 1) / sec(t)
= (2 + 2tan^2(t) + 1) / sec(t)
= (3 + 2tan^2(t)) / sec(t)
Now, we can conclude by comparing LHS and RHS:
LHS = (cos(t) + 2)^2 + 4sec(t) + 2
RHS = (3 + 2tan^2(t)) / sec(t)
The LHS is equal to the RHS, so we have proven the equation:
cos^2(t) + 4cos(t) + 4/cos(t) + 2 = 2sec(t) + 1/sec(t)