Chapter 7
Simplification of Trigonometric Expressions
An algebraic combination of the trig ratios my be named as a trigonometric expression. Often, converting a complicated trig expression into a simple one (by using trig identities) proves to be useful and rewarding. In doing so, you will be asked to verify a trig identity that has a somewhat complicated form on one side of its equal sign. It is better to generally keep one side untouched and work the other side until it becomes identical to the untouched side. Here are some examples:
Example 1:
The right side is left untouched. All conversions are done on the left side. Note that in the numerator the product (tana)(cota) is replaced by 1 because (tana)(cota) = 1. In the denominator, 1sin^{2}a = cos^{2}a because sin^{2}a + cos^{2}a = 1. Finally, (sina)/(cosa) is replaced by tana. 
Example 2:
cot^{2}b is replaced by
cos^{2}b/sin^{2}b,
tan^{2}b is replaced by sin^{2}b/cos^{2}b, and simplified. Then, cos^{2}b + sin^{2}b is replaced by 1, and finally, 1 + tan^{2}b is replaced by 1/cos^{2}b. 
Example 3:
1/sin^{2}b is replaced by
1+cot^{2}b, 1/cos^{2}b is replaced by 1+tan^{2}b, and 2 is replaced by 2·1 or 2·tanbcotb. Finally, the algebraic identity x^{2} + y^{2} +2xy = (x + y)^{2} is used to get the final result. 
Examples 4 and 5:
Examples 5: Verify that sin^{2}x  sin^{2}y = cos^{2}y  cos^{2}x. Solution: On the left side, replacing each sin^{2} by 1  cos^{2}, we get: 1  cos^{2}x ( 1 cos^{2}y) = 1  cos^{2}x  1 + cos^{2}y = cos^{2}y  sin^{2}x.

Examples 6: Determine a, b, and c such that the expression
(a+2b)x^{2} + (b  c)x + c = 16x +25 holds true for all values of x.
Solution: To make this equality true for all values of x, we have to force the coefficients of x^{2}, x^{1}, and x^{0} to become the same on both sides of the equality.
Note that x^{0} =1, and x^{1} = x. We write the given equality as
(a+2b)x^{2} + (b  c)x^{1} + cx^{0} = 0x^{2} +16x^{1} +25x^{0}.
Equating the corresponding coefficients, we must have:
a + 2b = 0,
bc = 16, and
c = 25.
Solving these 3 equations as a simultaneous set, we get :
c = 25, b = 41, and a = 82.
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Examples 7: Determine a, b, and c such that the expression
(a+b)sinx + (b  c)cosx + (c+b)tanx = 8cosx 6tanx does not depend on x.
Solution: To make this equality true for any value of x; in other words, to make it independent of x, we need to force the coefficients of sinx, cosx, and tanx to become the same on both sides of the equality. We write the given equality as
(a+b)sinx + (b  c)cosx + (c+b)tanx = (0)sinx +(8)cosx (6)tanx.
Equating the corresponding coefficients, we must have:
a + b = 0,
b  c = 8, and
c + b = 6.
Solving these 3 equations as a simultaneous set, we get :
a = 1, b = 1, and c = 7.
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Exercises: Verify the following identities:
Verify the following identities: