Claim: For all theta such that
-pie/2<theta<pie/2 the following holds true:
(1+tan(theta))^2=1/cos(theta)
LS = (1+ tanØ)^2
= 1 + 2tanØ + tan^2 Ø
= 1 + 2tanØ + sec^2 Ø - 1
= 2sinØ/cosØ + 1/cos^2 Ø
= (2sinØcosØ + 1)/cos^2 Ø
= (sin^2 Ø + cos^2 Ø + 2sinØcosØ)/cos^2 Ø
= (sinØ + cosØ)^2 /cosØ
≠ RS
The identity is false , all we need is one example when it does not work
e.g. Ø=30°
LS = (1+tan30°)^2 = appr 2.488
RS = 1/cos30° = 1.15
To verify the claim, we can start by manipulating the expression on the left side of the equation.
Given: (1 + tan(theta))^2 = 1/cos(theta)
Using the identity tan^2(theta) + 1 = sec^2(theta), we can rewrite the left side of the equation:
(1 + tan(theta))^2 = (1 + tan(theta))^2 * (1 + tan(theta))^0
= (1 + tan(theta))^2 * (sec^2(theta))^0
= (1 + tan(theta))^2 * (1/cos^2(theta))^0
= (1 + tan(theta))^2 * 1
= (1 + tan(theta))^2
Now, let's simplify the right side of the equation:
1/cos(theta) = cos^(-1)(theta)
= sec(theta)
So our equation becomes:
(1 + tan(theta))^2 = sec(theta)
Next, we'll use the identity sec(theta) = 1/cos(theta):
(1 + tan(theta))^2 = 1/cos(theta)
Now, we can square root both sides of the equation:
1 + tan(theta) = ±sqrt(1/cos(theta))
Squaring both sides again:
(1 + tan(theta))^2 = 1/cos(theta)
We've reached the original equation, so by substituting the right side back in, we have:
1/cos(theta) = 1/cos(theta)
Therefore, the claim is true for all theta satisfying -π/2 < theta < π/2.
To verify the claim, we need to simplify the expression (1 + tan(theta))^2 and compare it to 1/cos(theta) for all values of theta within the given range (-π/2 < theta < π/2). Let's go through the steps.
Step 1: Start with the expression (1 + tan(theta))^2.
(1 + tan(theta))^2 = (1 + sin(theta)/cos(theta))^2
Step 2: Expand the expression using the binomial square formula.
(1 + sin(theta)/cos(theta))^2 = (cos(theta) + sin(theta))^2 / cos^2(theta)
= cos^2(theta) + 2sin(theta)cos(theta) + sin^2(theta) / cos^2(theta)
Step 3: Apply the Pythagorean identity sin^2(theta) + cos^2(theta) = 1.
cos^2(theta) + 2sin(theta)cos(theta) + sin^2(theta) = 1 + 2sin(theta)cos(theta)
Step 4: Simplify the expression by dividing both sides by cos^2(theta).
(1 + tan(theta))^2 = (1 + 2sin(theta)cos(theta)) / cos^2(theta)
Step 5: Recall that tan(theta) = sin(theta) / cos(theta).
(1 + tan(theta))^2 = (1 + 2tan(theta)) / cos^2(theta)
Now, we need to compare the expression (1 + tan(theta))^2 to 1/cos(theta) for all theta values in the range -π/2 < theta < π/2.
Step 6: Simplify 1/cos(theta).
1/cos(theta) = sec(theta)
Step 7: Substitute sec(theta) for 1/cos(theta) in the expression (1 + tan(theta))^2 and simplify.
(1 + tan(theta))^2 = (1 + 2tan(theta)) / cos^2(theta) = (1 + 2tan(theta)) / sec^2(theta)
= (1 + 2tan(theta)) * cos^2(theta)
To conclude, the claim (1 + tan(theta))^2 = 1/cos(theta) is true for all theta such that -π/2 < theta < π/2, as the simplified expression (1 + tan(theta))^2 equals (1 + 2tan(theta)) * cos^2(theta), which is equal to 1/cos(theta) (sec(theta)).