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, 1-sin2a = cos2a because

sin2a + cos2a = 1.

Finally, (sina)/(cosa) is replaced by tana.

Example 2:

cot2b is replaced by cos2b/sin2b,

tan2b is replaced by sin2b/cos2b, and simplified.

Then, cos2b + sin2b is replaced by 1, and finally,

1 + tan2b is replaced by 1/cos2b.

Example 3:

1/sin2b is replaced by 1+cot2b,

1/cos2b is replaced by 1+tan2b, and

2 is replaced by 21 or 2tanbcotb.

Finally, the algebraic identity

x2 + y2 +2xy = (x + y)2 is used to get the final result.

Examples 4 and 5:

Examples 5:

Verify that

 sin2x - sin2y = cos2y - cos2x.

Solution: 

On the left side, replacing each sin2 by 1 - cos2, we get: 

1 - cos2x -( 1- cos2y) =

1 - cos2x - 1 + cos2y =

cos2y - sin2x.

 

 

Examples 6: Determine a, b, and c such that the expression

(a+2b)x2 + (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  x2, x1, and x0 to become the same on both sides of the equality. 

Note that  x0 =1,  and  x1 = x.  We write the given equality as

(a+2b)x2 + (b - c)x1  + cx0  =  0x2 +16x1 +25x0.

Equating the corresponding coefficients, we must have:

a + 2b = 0,

b-c = 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: