# Prove The Following - 2 (Trigonometry)

This note facilitates you to prove good Trigonometry questions. So please use this note. I will try my best to post these types of notes from now.

$$\color{green}\text{Prove the following Identities :}$$

1. $$(cosec A - sin A)(sec A - cos A)(tan A + cot A) = 1$$ $$\boxed{\text{proved in 1 method}}$$

2. $$\frac{sec A - tan A}{sec A + tan A} = 1 - 2sec A \cdot tan A + 2tan^2A$$

3. $$(sin A + cosec A)^2 + (cos A + sec A)^2 = 7 + tan^2A + cot^2A$$ $$\boxed{\text{proved in 1 method}}$$

4. $$sec^2A \cdot cosec^2A = tan^2A + cot^2A + 2$$

5. $$\frac{1}{1 + cos A} + \frac{1}{1 - cos A} = 2cosec^2A$$ $$\boxed{\text{proved in 1 method}}$$

6. $$\frac{sec A}{sec A + 1} + \frac{sec A}{sec A - 1} = 2 cosec^2A$$

7. $$\frac{1 + cos A}{1 - cos A} = \frac{tan^2A}{(sec A - 1)^2}$$

8. $$\frac{cot^2A}{(cosec A + 1)^2} = \frac{1 - sin A}{1 + sin A}$$

9. $$\frac{1 + sin A}{cos A} + \frac{cos A}{1 + sin A} = 2 sec A$$

10. $$\frac{1 - sin A}{1 + sin A} = (sec A - tan A)^2$$

If anybody proved any one of these I will mark that question as proved. But in Trigonometry one can prove a proof in many methods. So I will also mention in how many methods the proof is proved. Try your best to prove these. These are very easy if you apply little brain.

For more of these see my set Proof Based Notes

Note by Ram Mohith
3 weeks ago

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5) $$\frac{1}{1 + cos A} + \frac{1}{1 - cos A}$$

$$\frac{1 - cos A + 1 + cos A}{1 - cos^2A}$$

$$\frac{2}{sin^2A} = 2 \cdot cosec^2A$$

Hence Proved

- 2 weeks, 2 days ago

1) $$(cosec A - sin A)(sec A - cos A)(tan A + cot A) = 1$$

$$cosec A - sin A = \frac{1}{sin A} - sin A = \large\frac{1 - sin^2A}{sin A} = \frac{cos^2A}{sin A}$$

$$sec A - cos A = \frac{1}{cos A} - cos A = \large\frac{1 - cos^2A}{cos A} = \frac{sin^2A}{cos A}$$

$$tan A + cot A = \large\frac{sin A}{cos A} + \frac{cos A}{sin A} = \frac{sin^2A + cos^2A}{sin A \cdot cos A} = \frac{1}{sin A \cdot cos A}$$

$$LHS \implies \large\frac{cos^2A}{sin A} \times \frac{sin^2A}{cos A} \times \frac{1}{sin A \cdot cos A} = 1$$

$$\color{red}\text{Hence Proved}$$

- 2 weeks, 6 days ago

3) $$(sin A + cosec A)^2 + (cos A + sec A)^2 = 7 + tan^2A + cot^2A$$

$$LHS \implies (sin A + cosec A)^2 + (cos A + sec A)^2$$

$$\implies sin^2A + cosec^2A + 2 \cdot sin A \cdot cosec A + cos^2A + sec^2A + 2 \cdot cos A \cdot sec A$$

$$\implies sin^2A + cos^2A + 2 + cosec^2A + sec^2A + 2$$ (since $$sin A \cdot cosec A = cos A \cdot sec A = 1$$)

$$\implies 1 + 4 + 1 + tan^2A + 1 + cot^2A$$ (since $$cosec^2A = 1 + cot^2A$$ and $$sec^2A = 1 + tan^2A$$)

$$\implies 7 + tan^2A + cot^2A$$

$$\color{green}\text{Hence Proved}$$

- 3 weeks ago