Chapter in a nutshell: A trigonometric identity is true for all values of the angle. The three fundamental ones come from Pythagoras: $\sin^2\theta+\cos^2\theta=1$, $1+\tan^2\theta=\sec^2\theta$, $1+\cot^2\theta=\csc^2\theta$. Proving identities means converting one side to the other using these plus the ratio/reciprocal relations.
1. Key Idea
- An identity holds for every angle (unlike an equation, true only for some values).
- Strategy: convert everything to sin and cos, use the Pythagorean identities, and simplify one side to match the other.
2. Fundamental Identities (current chapter)
- $\sin^2\theta+\cos^2\theta=1$
- $1+\tan^2\theta=\sec^2\theta$ (i.e. $\sec^2\theta-\tan^2\theta=1$)
- $1+\cot^2\theta=\csc^2\theta$ (i.e. $\csc^2\theta-\cot^2\theta=1$)
- Useful rearrangements: $\sin^2\theta=1-\cos^2\theta$; $\tan^2\theta=\sec^2\theta-1$; etc.
3. Prerequisite Formulas (earlier classes — in full)
Ratios (right triangle): $\sin\theta=\dfrac{\text{opp}}{\text{hyp}}$, $\cos\theta=\dfrac{\text{adj}}{\text{hyp}}$, $\tan\theta=\dfrac{\text{opp}}{\text{adj}}$. Reciprocal ratios: $\csc\theta=\dfrac{1}{\sin\theta}$, $\sec\theta=\dfrac{1}{\cos\theta}$, $\cot\theta=\dfrac{1}{\tan\theta}$. Quotient relations: $\tan\theta=\dfrac{\sin\theta}{\cos\theta}$, $\cot\theta=\dfrac{\cos\theta}{\sin\theta}$. Complementary angles: $\sin(90^\circ-\theta)=\cos\theta$, $\cos(90^\circ-\theta)=\sin\theta$, $\tan(90^\circ-\theta)=\cot\theta$, $\sec(90^\circ-\theta)=\csc\theta$. Standard values:| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan | 0 | 1/√3 | 1 | √3 | ∞ |
4. Worked Example (one, for the method)
Q. Prove $\dfrac{\sin\theta}{1+\cos\theta}+\dfrac{1+\cos\theta}{\sin\theta}=2\csc\theta$. Solution: LHS $=\dfrac{\sin^2\theta+(1+\cos\theta)^2}{\sin\theta(1+\cos\theta)}=\dfrac{1+2\cos\theta+1}{\sin\theta(1+\cos\theta)}=\dfrac{2(1+\cos\theta)}{\sin\theta(1+\cos\theta)}=\dfrac{2}{\sin\theta}=2\csc\theta$. (used $\sin^2+\cos^2=1$).5. Common Mistakes to Avoid
- Writing $\sin^2\theta$ as $\sin\theta^2$ — it means $(\sin\theta)^2$.
- Using $1-\tan^2\theta=\sec^2\theta$ — it is $1+\tan^2\theta=\sec^2\theta$.
- Cross-multiplying both sides while "proving" — work one side to equal the other.
- Forgetting $\tan=\sin/\cos$ and the reciprocal relations when stuck.
6. Likely Exam Questions (with crisp answers)
- State the three fundamental identities. → $\sin^2+\cos^2=1$; $1+\tan^2=\sec^2$; $1+\cot^2=\csc^2$.
- Express $\sin^2\theta$ in terms of cos. → $1-\cos^2\theta$.
- What is $\sec^2\theta-\tan^2\theta$? → 1.
- Write $\tan\theta$ as a ratio of sin and cos. → $\sin\theta/\cos\theta$.
- Value of $\sin30^\circ$ and $\cos60^\circ$? → ½ and ½.
- Simplify $\sin(90^\circ-\theta)$. → $\cos\theta$.
- Reciprocal of $\cos\theta$? → $\sec\theta$.
- What is $\csc^2\theta-\cot^2\theta$? → 1.
- Value of $\tan45^\circ$? → 1.
- How does one prove an identity? → Simplify one side using known identities until it equals the other.
Extended & Prerequisite Formulas (full)
- $\sin^2+\cos^2=1$; $1+\tan^2=\sec^2$; $1+\cot^2=\csc^2$ (and rearrangements).
- Ratios sin=opp/hyp, cos=adj/hyp, tan=opp/adj; reciprocals csc,sec,cot; tan=sin/cos, cot=cos/sin.
- Complementary: sin(90−θ)=cosθ, tan(90−θ)=cotθ, sec(90−θ)=cscθ.
- Values: sin 0,½,1/√2,√3/2,1 (0–90°); cos reversed; tan 0,1/√3,1,√3,∞. Pythagoras hyp²=opp²+adj².