IDENTITIES:
1) Tan x = Sin x / Cos x
2) Sin^2 x + Cos^2 x = 1
Using these two identities; equations will be able to be manipulated to allow you so simplify / solve them.
For Quadratic Equations always remember:
1) Try and make the equation only (sin^2 x / sin x) or (cos^2 x / cos x). Eliminate all other terms using the identities above.
2) An equation can only be solved if sin x = k or cos x = k if 1 ≥ k ≥ -1.
Solutions to tan x = k exist for ALL VALUES of k.
Example 1:
"Show that Tan x + (1/Tan x) can be written in the form (1/Cos x Sin x)".
Tan x + (1/Tan x)
This is the original starting point given in the question.
(Sin x / Cos x) + (Cos x / Sin x)
Tan x = (Sin x / Cos x) and (1/ Tan x) = (Cos x / Sin x)
(Sin^2 x + Cos^2 x/Cos x Sin x)
Cross multiply and make the two fractions into one.
(1/Cos x Sin x)
Sin^2 x + Cos^2 x = 1 Therefore the top part of the bracket becomes one and the denominator stays the same.
Example 2:
"Show that 5cos x = 1 + 2Sin^2 x can be written in the form 2cos^2 x + 5Cos x -3 = 0".
5cos x = 1 + 2Sin^2 x
This is the original starting point given in the question.
5cos x = 1 + 2(1-Cos^2 x)
Sin^2 x + Cos^2 x = 1 REARRANGED = Sin^2 x = 1 - Cos^2 x
Sin^2 x has been substituted for 1 - Cos^2 x.
5cos x = 1 + 2 - 2 Cos^2 x
The brackets have been expanded from 2(1-Cos^2 x) to 2 - 2 Cos^2 x
2 Cos^2 x + 5Cos x -3 = 0
The terms have been grouped and formatted as positioned in the question.
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