If sec x = 17/8; verify that 3 - 4sin^2x /4cos^2x -3 = 3- tan^2x / 1-3tan^2x.. I have used x in place of theta. It isn't multiplication symbol.. ^2 means to the power 2. Pls help
secx = 17/8, so
cosx = 8/17
sinx = 15/17
tanx = 15/8
Now just plug in your values:
(3-4sin^2x)/(4cos^2x-3)
= (3-4(15/17)^2)/(4(8/17)^2-3)
= 33/611
(3-tan^2x)/(1-3tan^2x)
= (3-(15/8)^2)/(1-3(15/8)^2)
= 33/611
To verify the given equation:
3 - 4sin^2(x) / 4cos^2(x) - 3 = 3 - tan^2(x) / 1 - 3tan^2(x)
We can work on both sides separately and simplify using the given value sec(x) = 17/8.
Let's start with the left-hand side (LHS):
LHS = 3 - 4sin^2(x) / 4cos^2(x) - 3
Step 1: Simplify 4sin^2(x) / 4cos^2(x)
Since sec(x) = 1/cos(x), we can rewrite it as:
4sin^2(x) / (4(1 / sec^2(x)))
Step 2: Substitute sec(x) = 17/8
4sin^2(x) / (4(1 / (17/8)^2))
= 4sin^2(x) / (4(1 / (289/64)))
= 4sin^2(x) / (4/(289/64))
= 4sin^2(x) * (64/4 * 289)
= 64sin^2(x) / 289
Now, replacing it back into the LHS equation:
LHS = 3 - 64sin^2(x) / 289
Next, let's simplify the right-hand side (RHS):
RHS = 3 - tan^2(x) / 1 - 3tan^2(x)
Step 1: Simplify tan^2(x) / 1 - 3tan^2(x)
Since sec(x) = 1/cos(x), we can rewrite it as:
tan^2(x) / (1 - 3(1 / sec^2(x)))
Step 2: Substitute sec(x) = 17/8
tan^2(x) / (1 - 3(1 / (17/8)^2))
= tan^2(x) / (1 - 3(1 / (289/64)))
= tan^2(x) / (1 - 3 / (289/64))
= tan^2(x) * (64/289 * 64/61)
= 64tan^2(x) / 61
Now, replacing it back into the RHS equation:
RHS = 3 - 64tan^2(x) / 61
Comparing LHS and RHS:
LHS = 3 - 64sin^2(x) / 289
RHS = 3 - 64tan^2(x) / 61
We can see that LHS and RHS are not equal. Hence, the given equation is not satisfied when sec(x) = 17/8.