Simplifying before you substitute means using an identity or a division to reduce an expression to the one ratio you have been given. It saves time and avoids square roots. This skill appears when a question gives tan x, sin x or cos x and asks for the value of a longer expression.
The lesson builds on proving identities and belongs to trigonometric identities.
Why is this shorter than the long route?
Suppose tan x = 3 and you want (2 sin x + cos x) / (sin x − cos x). The long route finds sin x = 3/√10 and cos x = 1/√10, then substitutes into the expression and clears surds.
The short route notices that every term has one sine or cosine. Dividing the numerator and denominator by cos x gives an expression in tan x alone, and there is nothing left to work out.
How do you simplify first?
- List what you are given and what you need. For example: given tan x, need a mixed expression.
- Choose the tool. Divide by cos x to bring in tan x. Use sin²x + cos²x = 1 to reduce squares. Use 1 + tan²x = sec²x to bring in sec.
- Divide or replace every term. Do not forget the denominator.
- Substitute the given value into the simplified expression.
- Check by testing with a right-angled triangle, if time allows.
Worked example
Given that tan x = 3, find the value of (2 sin x + cos x) / (sin x − cos x).
Step 1, choose the tool: the expression has one sine or cosine per term, so divide top and bottom by cos x.
Step 2, divide: (2 sin x / cos x + cos x / cos x) / (sin x / cos x − cos x / cos x) = (2 tan x + 1) / (tan x − 1).
Step 3, substitute: (2 × 3 + 1) / (3 − 1) = 7 / 2.
Answer: 7/2.
Check with a triangle: tan x = 3 gives opposite 3, adjacent 1, hypotenuse √10. Then sin x = 3/√10 and cos x = 1/√10, so the expression is (6/√10 + 1/√10) / (3/√10 − 1/√10) = 7/2. Both methods agree.
The mistake to watch for
A common slip is to read tan x = 3 as sin x = 3 and cos x = 1.
Mistaken working: tan x = 3, so sin x = 3 and cos x = 1. Then sin x cos x = 3 × 1 = 3.
The student used the ratio as if it were the length of the side.
For the worked example above this slip gives the right answer by luck, because every term has the same degree.
For sin x cos x it gives 3, which is impossible: sin x cos x cannot be more than 1/2. The correct route is sin x cos x = tan x cos²x = tan x / (1 + tan²x) = 3/10, because 1/cos²x = 1 + tan²x. The triangle check gives (3/√10)(1/√10) = 3/10.
Check yourself
1. Given tan x = 1/2, find (sin x − cos x) / (sin x + cos x).
Show answer
Divide by cos x: (tan x − 1) / (tan x + 1) = (1/2 − 1) / (1/2 + 1) = (−1/2) / (3/2) = −1/3.
2. Simplify sin x / (1 − cos²x).
Show answer
1 − cos²x = sin²x, so the expression is sin x / sin²x = 1 / sin x = cosec x (for sin x ≠ 0).
3. Given sin x = 0.6 and x is obtuse, find cos x and tan x.
Show answer
cos²x = 1 − 0.36 = 0.64, so cos x = ±0.8. An obtuse angle has a negative cosine, so cos x = −0.8. Then tan x = 0.6 / (−0.8) = −0.75.
Where this leads next
Next, state excluded values in a trig identity, because dividing by cos x in this lesson quietly assumed cos x ≠ 0. The non-calculator working trainer and the quadratic structure explorer are useful for the algebra.
If you often choose the longer method first, our teachers show how to scan a question for the shortcut in online one-to-one Additional Mathematics tuition.