Class 10 Trigonometry Most Important Questions

Trigonometry Most Important Questions

Q1. if  `sec^2\theta\left(1-\sin\theta\right)\left(1+\sin\theta\right)=K`  then find the value of K.

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Q2. Prove the following Identities:

i) `\left(\cos ec^2A-1\right)\left(secA+1\right)\left(secA-1\right)=1`  

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ii)  `1+\left(1+\cos A\right)\left(1-\cos A\right)\left(1+cot^2A\right)=2`  

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iii)  `\tan^2A+\left(1+\tan^2A\right)\left(1+\sin A\right)\left(1-\sin A\right)=sec^2A`   

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iv)  `cot\theta-\tan\theta=\frac{2\cos^2\theta-1}{\sin\theta\cos\theta}`  

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v) `\frac{\left(1+\sin\theta\right)^2+\left(1-\sin\theta\right)^2}{2\cos^2\theta}=\frac{1+\sin^2\theta}{1-\sin^2\theta}`    

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vi) `\frac{\cos^2\theta}{\sin\theta}-\cos ec\theta+\sin\theta=0`   

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Q3. Prove the following Identities:

i)  `\tan^4\theta+\tan^2\theta=sec^4\theta-sec^2\theta`

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ii)  `\sin^2\theta+\cos^4\theta=\cos^2\theta-\sin^4\theta`

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iii)   `csc^4\theta-csc^2\theta=cot^4\theta+cot^2\theta`

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iv)  `\sin^6\theta+\cos^6\theta=1-3\sin^2\theta\cos^2\theta`

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Q4. Prove that:    `1+\frac{cot^{2\}A}{1+\cos ecA}=cosecA`

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Q5. Prove that:   `\left(\sin\alpha\+\cos\alpha\right)\left(\tan\alpha\+\cot\alpha\right)=\sec\alpha\+\cosec\alpha`

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Q6. Prove that:   `\frac{\sin^3\theta-\cos^3\theta}{\sin\theta\-\cos\theta}\-\sin\theta\cos\theta\=1`

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Q7. Prove that:   `sec\theta\left(1-\sin\theta\right)\left(sec\theta+\tan\theta\right)\=1`

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Q8. Prove that:     `\sin\theta\left(1+\tan\theta\right)\+\cos\theta\left(1+cot\theta\right)\=sec\theta\+\cosec\theta`

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Q9. Prove that:   `\left(1+cot\A\-\cosec\A\right)\left(1+\tan\A\+\sec\A\right)\=\2`

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Q10. Prove that:   `\left(1-\sin\theta\+\cos\theta\right)^2\=\2\left(1+\cos\theta\right)\left(1-\sin\theta\right)`

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Q11. Prove that: `\sqrt{sec^2\theta+\cosec^2\theta}=\tan\theta+cot\theta`

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Q12. Prove that:  `\frac{\cosecA}{\cos ecA-1}+\frac{\cosecA}{\cos ecA+1}=2+2\tan^2A\\`

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Q13. Prove that:  `\left(1+\frac1{\tan^2\theta}\right)\left(1+\frac1{cot^2\theta}\right)=\frac1{\sin^2\theta-\sin^4\theta}\\`

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Q14. Prove that:  `\frac{\cos\theta}{1-\tan\theta}+\frac{\sin^2\theta}{\sin\theta-\cos\theta}=\sin\theta+\cos\theta\\`

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Q15. Prove that:  `\frac{\cos^2\theta}{1-\tan\theta}+\frac{\sin^3\theta}{\sin\theta-\cos\theta}=1+\sin\theta\cos\theta\\`

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Q16. Prove that:  `\frac{cotA+\cos ecA-1}{cotA-\cos ecA+1}=\frac{1+\cos A}{\sin A}\\`

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Q17. Prove that:  `\frac{1+secA-\tan A}{1+secA+\tan A}=\frac{1-\sin A}{\cos A}\\`

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Q18. Prove that:  `\frac{\tan A+\tan B}{cotA+cotB}=\tan A\\tan B\\`

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Q19. Prove that:  `\frac{cotA+\tan B}{cotB+\tan A}=cotA\tan B\\`

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Q20. Prove that:  `\frac{secA-1}{secA+1}=\frac{\sin^2A}{\left(1+\cos A\right)^2}\\`

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Q21. Prove that

Q22. Prove that

Q23. Prove that

Q24. Prove that

Q25. Prove that

Q26. Prove that

Q27. Prove that

Q28. Prove that

Q29. Prove that `${\csc ^4}\theta  - {\csc ^2}\theta  = {\cot ^4}\theta  + {\cot ^2}\theta $`

Q30. Prove that `\sqrt \frac\csc A - 1\csc A + 1  + \sqrt {\frac{{\csc A + 1}}{{\csc A - 1}}}  = 2\sec A`

Module 8.1

Q1. In each of the following, one of the six trigonometric ratios is given. Find the values of the other trigonometric ratios.

a)     sinθ=23$\sin �=\frac{2}{3}$ 

b)     cosθ=45$\cos �=\frac{4}{5}$ 

c)     tanθ=512$\tan �=\frac{5}{12}$ 

Q2. In a  , right angled at B, AB = 24 cm, BC = 7 cm. Determine  i) sin A,  cos A    ii) sin C, cos C

Q3. In ΔPQR$����$, right angled at Q, PQ = 4 cm and RQ = 3 cm. Find the values of sin P, sin R, sec P and sec R.

Q4. If secA=10−−√$\mathrm{secA}=\sqrt{10}$ , find the other trigonometric ratios.

Q5. If sinθ=35$\sin �=\frac{3}{5}$ , Show that 4tanθ+3sin−6cosθ=0$4\tan �+3\sin -6\cos �=0$ .

Q6. In right triangle ABC$���$ , right angle at B$�$ , AB=5 cm$��=5��$ and AC−BC=1 cm$��-��=1��$ determine the value of sinA, cosA,$\sin �,\cos �,$ and tanA$\tan �$.

Q7. If 5secθ=13$5\sec �=13$ , Show that 2sinθ−3cosθ4sinθ−9cosθ=3$\frac{2\sin �-3\cos �}{4\sin �-9\cos �}=3$ .

Q8. If tanθ=17√$\tan �=\frac{1}{\sqrt{7}}$ , Show that cosec2θ−sec2θcosec2θ+sec2θ=34$\frac{\cos {\text{ec}}^2�-{\sec }^2�}{\cos {\text{ec}}^2�+{\sec }^2�}=\frac{3}{4}$ 

Q9. If 20cosθ=12$20\cos �=12$ , find the value of sinθ−1tanθ2tanθ$\frac{\sin �-\frac{1}{\tan �}}{2\tan �}$ 

Q10. If 3cosθ−4sinθ=2cosθ+sinθ$3\cos �-4\sin �=2\cos �+\sin �$ find tanθ$\tan �$ 

Q11. If 15tanθ=14$15\tan �=14$ , evaluate 2sinθcosθcos2θ−sin2θ$\frac{2\sin �\cos �}{{\cos }^2�-{\sin }^2�}$ 

Q12. If cotθ=1312$\cot �=\frac{13}{12}$ evaluate cot2θ−1cosθ×1cosecθ$\frac{{\cot }^2�-1}{\cos �}\times \frac{1}{\cos ���}$ 

Q13. Given that 16tanA=12$16\tan �=12$, find the value of cosA+sinAcosA−sinA$\frac{\cos �+\sin �}{\cos �-\sin �}$ 

Q14.  If cotθ=13√$\cot �=\frac{1}{\sqrt{3}}$, Show that sin2θ−cos2θ2cos2θ−sin2θ=−2$\frac{{\sin }^2�-{\cos }^2�}{2{\cos }^2�-{\sin }^2�}=-2$.

Q15. If 3–√tanθ=1$\sqrt{3}\tan �=1$, then find the value of sin2θ−cos2θ${\sin }^2�-{\cos }^2�$ 

Q16. If cosecA=2$\cos \text{ec}�=2$, find the value of cotA+sinA1+cosA$\cot �+\frac{\sin �}{1+\cos �}$     

Q17. If tanA=2–√−1,$\tan �=\sqrt{2}-1,$ Show that sinA⋅cosA=2√4$\sin �\cdot \cos �=\frac{\sqrt{2}}{4}$ 

Q18. If sinθ=a2−b2a2+b2$\sin �=\frac{{�}^2-{�}^2}{{�}^2+{�}^2}$ , find the value of other trigonometric ratios.

Q19. If  sinθ=13$\sin �=\frac{1}{3}$ , evaluate cosA.cosecA+tanAsecA$\cos �.\cos \text{ec}�+\tan �\sec �$.

Q20. If sinθ=35$\sin �=\frac{3}{5}$, find the value of (tanθ+secθ)2${\left(\tan �+\sec �\right)}^2$ 

Q21. If cosecA=2$\cos \text{ec}�=2$, find the value of 1tanA+sinA1+cosA$\frac{1}{\tan �}+\frac{\sin �}{1+\cos �}$ 

Q22. If sinα=12$\sin �=\frac{1}{2}$, show that (3cosα−4cos3α)=0$\left(3\cos �-4{\cos }^3�\right)=0$ .

Q23. If 3tanθ=4$3\tan �=4$ , find the value of 5sinθ−3cosθ5sinθ+2cosθ$\frac{5\sin �-3\cos �}{5\sin �+2\cos �}$ .

Q24. If tanA=1$\tan �=1$ and tanB=3–√$\tan �=\sqrt{3}$, find the value of cosA.cosB−sinA.sinB$\cos �.\cos �-\sin �.\sin �$.