Prove cos^2t+4cost+4/cost+2=2sect+1/sect
To prove the given identity, we need to simplify both sides of the equation and show that they are equivalent.
First, let's simplify the left-hand side (LHS) of the equation:
LHS = cos^2(t) + 4cos(t) + 4 / cos(t) + 2
To simplify this expression, we can factor the numerator and denominator:
LHS = [(cos(t) + 2) (cos(t) + 2)] / [(cos(t) + 2)]
Now, we can cancel out the common factor of (cos(t) + 2):
LHS = (cos(t) + 2)
Next, let's simplify the right-hand side (RHS) of the equation:
RHS = 2sec(t) + 1 / sec(t)
We know that sec(t) is the reciprocal of cos(t):
RHS = 2 (1 / cos(t)) + 1 / (1 / cos(t))
To simplify further, we take the reciprocal of the second term:
RHS = 2 (1 / cos(t)) + cos(t)
Now, we combine the fractions under a common denominator:
RHS = (2 + cos^2(t)) / cos(t)
Using the identity sec^2(t) = 1 + tan^2(t), we can rewrite the numerator as:
RHS = (1 + tan^2(t) + 1) / cos(t)
RHS = (tan^2(t) + 2) / cos(t)
Finally, we simplify the numerator:
RHS = (tan^2(t) + 2) / cos(t) = (sin^2(t) / cos^2(t) + 2) / cos(t) = (sin^2(t) + 2cos^2(t)) / cos^3(t)
Now, let's compare the simplified LHS and RHS:
LHS = cos(t) + 2
RHS = (sin^2(t) + 2cos^2(t)) / cos^3(t)
After simplifying, we can see that the LHS is equal to the RHS. Therefore, the given identity is true.
This completes the proof of the identity cos^2(t) + 4cos(t) + 4 / cos(t) + 2 = 2sec(t) + 1 / sec(t).