Question

If sin θ + cos θ = √3, then prove that tan θ + cot θ =1
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Answer 1

$\huge{\underline{\underline{\mathfrak{Answer \colon}}}}$

Given,

$\large{ \sf{sin \: x + cos \: x = \sqrt{3} }}............(1)$

To Prove

$\sf{tan \: x + cot \: x = 1}$

Squaring Equation (1) on both sides,

$\large{ \sf{ {(sin \: x + \: cos \: x)}^{2}}} = \sqrt{3 {}^{2} } \\ \\ \large{ \leadsto \: \sf{sin {}^{2}x + cos {}^{2}x + 2sin \: x.cos \: x = 3 }} \\ \\ \large{ \leadsto \: \sf{1 + 2sin \: x.cos \: x = 3}} \\ \\ \huge{ \leadsto \: \boxed{ \sf{sin \: x.cos \: x = 1}}}$

Now,

$\large{ \sf{tan \: x + cot \: x}} \\ \\ \large{\rightarrow \: \sf{ \frac{sin \: x}{cos \: x} + \frac{cos \: x}{sin \: x} }} \\ \\ \large{ \rightarrow \: \sf{ \frac{sin {}^{2} x + cos {}^{2}x }{sin \: x.cos \: x} }} \\ \\ \huge{ \rightarrow \: \tt{1}}$

Henceforth,Proved

Answer 2

Answer:

answer in attachment

Step-by-step explanation:

sin θ + cos θ = √3, (given)

Squaring both sides,

we get

sin^2 θ +cos^2 θ +2cos θ sin θ =3

2sin θ cos θ =2

sin θ cos θ =1..............(i)

Now tan θ +cot θ =1

=sin θ /cos θ +cos θ /sin θ =1

= (sin^2 θ +cos ^2θ )/sin θ cos θ = 1/(sin θ cos θ )

=1/1 = 1

thank you!