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For angles a,b,c∈R with a+b+c=π, prove the following identities: tan^(−1) ((a/2))+tan^(−1) ((b/2))+tan^(−1) ((c/2))=(tan((a/2))tan((b/2))tan((c/2)))^(−1) Help

$${For}\:{angles}\:{a},{b},{c}\in\mathbb{R}\:{with}\:{a}+{b}+{c}=\pi,\: \\ $$$${prove}\:{the}\:{following}\:{identities}: \\ $$$$\mathrm{tan}^{−\mathrm{1}} \left(\frac{{a}}{\mathrm{2}}\right)+\mathrm{tan}^{−\mathrm{1}} \left(\frac{{b}}{\mathrm{2}}\right)+\mathrm{tan}^{−\mathrm{1}} \left(\frac{{c}}{\mathrm{2}}\right)=\left(\mathrm{tan}\left(\frac{{a}}{\mathrm{2}}\right)\mathrm{tan}\left(\frac{{b}}{\mathrm{2}}\right)\mathrm{tan}\left(\frac{{c}}{\mathrm{2}}\right)\right)^{−\mathrm{1}} \\ $$$$\mathrm{H}{elp} \\ $$$$ \\ $$

Question Number 115380 Answers: 2 Comments: 0

if a^2 +2ab+3b^2 =1,prove that (a+3b)^3 (d^2 b/da^2 )+2(a^2 +2ab+3b^2 )=0

$${if}\: \\ $$$${a}^{\mathrm{2}} +\mathrm{2}{ab}+\mathrm{3}{b}^{\mathrm{2}} =\mathrm{1},{prove}\:{that}\: \\ $$$$\left({a}+\mathrm{3}{b}\right)^{\mathrm{3}} \frac{{d}^{\mathrm{2}} {b}}{{da}^{\mathrm{2}} }+\mathrm{2}\left({a}^{\mathrm{2}} +\mathrm{2}{ab}+\mathrm{3}{b}^{\mathrm{2}} \right)=\mathrm{0} \\ $$

Question Number 115368 Answers: 1 Comments: 0

an open rectanqular container is to have a volume of 62.5cm^3 .find the least possible surface area of the material required

$${an}\:{open}\:{rectanqular}\:{container}\:{is}\:{to} \\ $$$${have}\:{a}\:{volume}\:{of}\:\mathrm{62}.\mathrm{5}{cm}^{\mathrm{3}} .{find}\:{the}\:{least} \\ $$$${possible}\:{surface}\:{area}\:{of}\:{the}\:{material} \\ $$$${required} \\ $$

Question Number 115367 Answers: 1 Comments: 0

solve xy^(′′) −(x^2 +1)y^′ =x^2 sin(2x)

$${solve}\:{xy}^{''} −\left({x}^{\mathrm{2}} +\mathrm{1}\right){y}^{'} \:\:={x}^{\mathrm{2}} {sin}\left(\mathrm{2}{x}\right) \\ $$

Question Number 115366 Answers: 2 Comments: 0

calculate ∫_(−1) ^2 (dx/(ch^2 x +sh^2 x))

$${calculate}\:\int_{−\mathrm{1}} ^{\mathrm{2}} \:\frac{{dx}}{{ch}^{\mathrm{2}} {x}\:+{sh}^{\mathrm{2}} {x}} \\ $$

Question Number 115365 Answers: 0 Comments: 0

calculate ∫∫_([0,1]^2 ) ((arctan(xy))/( (√(x^2 +y^2 ))))dxdy

$${calculate}\:\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{2}} } \:\:\frac{{arctan}\left({xy}\right)}{\:\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }}{dxdy} \\ $$

Question Number 115364 Answers: 1 Comments: 0

calculate ∫∫_([0,1]^2 ) (√(xy))(x^2 +y^2 )dxdy

$${calculate}\:\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{2}} } \:\:\sqrt{{xy}}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right){dxdy} \\ $$

Question Number 115363 Answers: 1 Comments: 0

evaluate ∫_0 ^(π/3) (1/(sin^2 x+cos^2 x))dx

$${evaluate} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{3}}} \frac{\mathrm{1}}{\mathrm{sin}^{\mathrm{2}} {x}+\mathrm{cos}^{\mathrm{2}} {x}}{dx} \\ $$

Question Number 115362 Answers: 1 Comments: 0

calculate ∫_0 ^1 ((sinx)/(1+x^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{sinx}}{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 115361 Answers: 1 Comments: 0

find ∫_0 ^∞ ((cos(πx^2 ))/((x^2 +3)^2 ))dx

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left(\pi{x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} +\mathrm{3}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 115348 Answers: 1 Comments: 0

If x ∈ (0,(π/2)) and 2cos x(sin x+cos x)+tan^2 x < sec^2 x has solution set is a<x<b. find the value of a+b

$${If}\:{x}\:\in\:\left(\mathrm{0},\frac{\pi}{\mathrm{2}}\right)\:{and}\:\mathrm{2cos}\:{x}\left(\mathrm{sin}\:{x}+\mathrm{cos}\:{x}\right)+\mathrm{tan}\:^{\mathrm{2}} {x}\:<\:\mathrm{sec}\:^{\mathrm{2}} {x}\: \\ $$$${has}\:{solution}\:{set}\:{is}\:{a}<{x}<{b}.\:{find}\:{the} \\ $$$${value}\:{of}\:{a}+{b} \\ $$

Question Number 115345 Answers: 3 Comments: 0

sec θ (sec θ (sin^2 θ)+2(√3) sin θ)=1 has the roots are θ_1 and θ_2 . Find the value of tan θ_1 ×tan θ_2 .

$$\mathrm{sec}\:\theta\:\left(\mathrm{sec}\:\theta\:\left(\mathrm{sin}\:^{\mathrm{2}} \theta\right)+\mathrm{2}\sqrt{\mathrm{3}}\:\mathrm{sin}\:\theta\right)=\mathrm{1} \\ $$$${has}\:{the}\:{roots}\:{are}\:\theta_{\mathrm{1}} \:{and}\:\theta_{\mathrm{2}} .\:{Find}\:{the} \\ $$$${value}\:{of}\:\mathrm{tan}\:\theta_{\mathrm{1}} ×\mathrm{tan}\:\theta_{\mathrm{2}} . \\ $$

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