Question

if tan² theta + cot² theta = 2, theta is an acute angle then tan³ theta + cot³ theta is equal to​

Answer 1

Answer:

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Step-by-step explanation:

$so \: tan {}^{2} θ + cot {}^{2} θ = 2 \\ so \: here \: tan {}^{2} θ + cot {}^{2} θ \: is \: in \: form \: a {}^{2} + b {}^{2} \\ so \: we \: know \: that \\ a {}^{2} + b {}^{2} = (a + b) {}^{2} - 2ab \\ hence \: accordingly \\ \\ tan {}^{2} θ + cot {}^{2} θ = (tanθ + cotθ) {}^{2} - 2.tanθ.cotθ \\ 2 = (tanθ + cotθ) {}^{2} - (2 \times 1) \\ since \: \: tanθ.cotθ = 1 \\ (tanθ + cotθ) {}^{2} = 2 + 2 \\ = 4 \\ taking \: square \: roots \: on \: both \: sides \\ we \: get \\ tanθ + cotθ = 2$

$now \: cubing \: both \: sides \\ we \: get \\ (tanθ + cotθ) {}^{3} = tan {}^{3} θ + cot {}^{3} θ \\ (2) {}^{3} = tan {}^{3} θ + cot {}^{3} θ + 3.tanθ.cotθ(tanθ + cotθ) \\ 8 = tan {}^{3} θ + cot {}^{3} θ + 3 \times 1 \times (2) \\ 8 = tan {}^{3} θ + cot {}^{3} θ + 6 \\ tan {}^{3} θ + cot {}^{3} θ = 8 - 6 = 2 \\ \\ hence \: tan {}^{3} θ + cot {}^{3} θ = 2$

Answer 2

Answer:

Step-by-step explanation: