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Prove that for real x, (tan(x+alpha))/(t...

Prove that for real x, `(tan(x+alpha))/(tan(x-alpha))` where `alpha ne pi/4, 3pi/4` cannot lies between `(1- sin2 alpha)/(1+ sin 2alpha)` and `(1+ sin2 alpha)/(1- sin 2alpha)`.

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A DAS GUPTA-Circular Functions, Identities -Exercise
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  3. Prove that for real x, (tan(x+alpha))/(tan(x-alpha)) where alpha ne pi...

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  4. Show that (sec^(2)theta - tan theta)/(sec^(2)+ tan theta) lies betwe...

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  5. Prove that tan alpha + 2tan 2alpha + 4tan 4alpha + 8cot 8alpha = cot ...

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  6. Find the sum: tan x + 1/2 tan x/2+ 1/2^(2) tan x/2^(2) +.....+1/2^(n-...

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  7. Prove that cosec 2alpha = cot alpha - cot 2alpha. Hence or otherwise ...

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  8. Prove that sum(r=1)^(10) = sin^(2) rpi/18 =5.

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  9. Sum the series: sin alpha - sin 2alpha + sin 3alpha - sin 4alpha + .....

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  10. Find sum(r=1)^(10) cos^(3) rpi/3.

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  11. Sum: sqrt(1 + sin 2alpha) + sqrt(1 + sin 4alpha) + sqrt(1 + sin 6alpha...

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  12. Prove that sin^(2) 12^circ + sin^(2)21^circ + sin^(2)39^circ + sin^(2)...

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  13. If A + B + C = pi, prove that cos A cosec B cosec C + cos B cosec C c...

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  14. Prove that in a triangle ABC, sin^(2)A - sin^(2)B + sin^(2)C = 2sin A...

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  15. Prove that in a triangle ABC, sum (cot B+ cotC)/(tan B + tan C) =1.

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  16. If A +B+C = 180^circ, prove that cos A/2+ cos B/2 +cos C/2 = 4cos (p...

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  17. If alpha + beta- gamma= pi, prove that sin^(2)alpha + sin^(2)beta - s...

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  18. If alpha + beta + gamma = 2 theta, prove that cos theta + cos(theta- ...

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  19. If A + B + C = pi, show that cot A + sinA/(sin B sin C) = cot B + sin...

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  20. If A + B + C = 0, prove that sum sin 2A = 2sum sin A * (1 + sum cos A...

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