Many trigonometric identities are difficult because the algebra around the trigonometric functions is not organised. Factorisation, common denominators and substitution often create the form in which a known identity becomes useful.
1. Factor before replacing functions
In 1−cos²x, the structure is already a difference of squares and also a Pythagorean identity. In expressions such as sin x−sin x cos²x, factor first:
sin x(1−cos²x)=sin x·sin²x=sin³x.
2. Common factors can expose cancellations
Consider sin²x+sin x cos x. Factor sin x:
sin x(sin x+cos x).
If this appears over sin x in a domain where sin x≠0, the common factor can then be cancelled legitimately.
3. Common denominators are often the main move
Simplify 1/(1−cos x)+1/(1+cos x).
The common denominator is (1−cos x)(1+cos x)=1−cos²x=sin²x.
The numerator becomes 2, so the expression is 2/sin²x=2cosec²x, where defined.
4. Convert only what helps
Replacing every trig function with sine and cosine is valid in many settings but can make an expression longer. Convert strategically.
For sec x tan x, converting both gives sin x/cos²x. That may be useful if the target is expressed using sine and cosine, but unnecessary if the target already contains sec and tan.
5. Substitution can reveal algebraic structure
If an expression contains repeated tan²x, temporarily let u=tan²x. Then sec²x=1+u. This can turn a trig simplification into ordinary algebra before translating back.
The substitution is a bookkeeping device, not a new identity.
6. Worked proof using a common denominator
Show that 1/(1−sin x)+1/(1+sin x)=2sec²x.
Combine:
[(1+sin x)+(1−sin x)]/[(1−sin x)(1+sin x)] = 2/(1−sin²x).
Since 1−sin²x=cos²x, the result is 2/cos²x=2sec²x.
7. Worked proof using factorisation
Show that (sin x+cos x)²−1=2sin x cos x.
Expand the square:
sin²x+2sin x cos x+cos²x−1.
Use sin²x+cos²x=1. The remaining expression is 2sin x cos x.
8. Rationalising-style multiplication can help
Expressions containing 1±sin x or 1±cos x sometimes simplify when multiplied by the conjugate-like partner. For example, (1−sin x)(1+sin x)=cos²x.
The aim is to create a Pythagorean form, not to copy a surd method blindly.
9. Common mistakes
- Cancelling across addition instead of factoring first.
- Using an incorrect common denominator.
- Converting every function and making the expression harder.
- Forgetting denominator restrictions after cancellation.
10. Practice
- Simplify sin x−sin x cos²x.
- Simplify 1/(1−cos x)+1/(1+cos x).
- Show that (sin x−cos x)²=1−2sin x cos x.
- Factor sin²x−sin x cos x.
- Explain why (sin x+1)/sin x cannot be simplified by cancelling sin x.
11. Answers
- sin³x.
- 2cosec²x.
- Expand and use sin²x+cos²x=1.
- sin x(sin x−cos x).
- sin x is not a common factor of the entire numerator.
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