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Question Number 105285 by bemath last updated on 27/Jul/20

simplify ((sin θ)/(1+cot θ)) + ((cos θ)/(1+ tan θ)) ?

$${simplify}\:\frac{\mathrm{sin}\:\theta}{\mathrm{1}+\mathrm{cot}\:\theta}\:+\:\frac{\mathrm{cos}\:\theta}{\mathrm{1}+\:\mathrm{tan}\:\theta}\:? \\ $$

Commented by bemath last updated on 27/Jul/20

thank you

$${thank}\:{you} \\ $$

Commented by som(math1967) last updated on 27/Jul/20

((sinθ)/(1+((cosθ)/(sinθ)))) +((cosθ)/(1+((sinθ)/(cosθ)))) =((sin^2 θ)/(sinθ+cosθ)) +((cos^2 θ)/(cosθ+sinθ)) =(1/(sinθ+cosθ)) [∵sin^2 θ+cos^2 θ=1]

$$\frac{\mathrm{sin}\theta}{\mathrm{1}+\frac{\mathrm{cos}\theta}{\mathrm{sin}\theta}}\:+\frac{\mathrm{cos}\theta}{\mathrm{1}+\frac{\mathrm{sin}\theta}{\mathrm{cos}\theta}} \\ $$$$=\frac{\mathrm{sin}^{\mathrm{2}} \theta}{\mathrm{sin}\theta+\mathrm{cos}\theta}\:+\frac{\mathrm{cos}^{\mathrm{2}} \theta}{\mathrm{cos}\theta+\mathrm{sin}\theta} \\ $$$$=\frac{\mathrm{1}}{\mathrm{sin}\theta+\mathrm{cos}\theta}\:\:\left[\because\mathrm{sin}^{\mathrm{2}} \theta+\mathrm{cos}^{\mathrm{2}} \theta=\mathrm{1}\right] \\ $$

Answered by bobhans last updated on 27/Jul/20

⇔ ((sin^2 x)/(sin x+cos x)) + ((cos^2 )/(cos x+sin x)) = (1/(sin x+cos x)) = (1/((√2) cos (x−(π/4)))) = ((sec (x−(π/4)))/(√2)) ▷

$$\Leftrightarrow\:\frac{\mathrm{sin}\:^{\mathrm{2}} {x}}{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}}\:+\:\frac{\mathrm{cos}\:^{\mathrm{2}} }{\mathrm{cos}\:{x}+\mathrm{sin}\:{x}}\:=\:\frac{\mathrm{1}}{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}} \\ $$$$=\:\frac{\mathrm{1}}{\sqrt{\mathrm{2}}\:\mathrm{cos}\:\left({x}−\frac{\pi}{\mathrm{4}}\right)}\:=\:\frac{\mathrm{sec}\:\left({x}−\frac{\pi}{\mathrm{4}}\right)}{\sqrt{\mathrm{2}}}\:\triangleright \\ $$

Commented by malwaan last updated on 27/Jul/20

(1/(sin x + cos x)) =(1/((√2)(sinx×(1/(√2))+cosx×(1/(√2))))) =(1/((√2) (sin x cos(π/4) + cos x sin(π/4)))) = (1/((√2) sin (x+(π/4))))=((cosec(x+(π/4)))/(√2))

$$\frac{\mathrm{1}}{{sin}\:{x}\:+\:{cos}\:{x}}\:=\frac{\mathrm{1}}{\sqrt{\mathrm{2}}\left({sinx}×\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}+{cosx}×\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\right)} \\ $$$$=\frac{\mathrm{1}}{\sqrt{\mathrm{2}}\:\left({sin}\:{x}\:{cos}\frac{\pi}{\mathrm{4}}\:+\:{cos}\:{x}\:{sin}\frac{\pi}{\mathrm{4}}\right)} \\ $$$$=\:\frac{\mathrm{1}}{\sqrt{\mathrm{2}}\:{sin}\:\left({x}+\frac{\pi}{\mathrm{4}}\right)}=\frac{{cosec}\left({x}+\frac{\pi}{\mathrm{4}}\right)}{\sqrt{\mathrm{2}}} \\ $$

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