Question

if sin theta + cos theta =√2 then prove that tan theta +cot
theta =2​

Answer 1

Answer:

Consider the given equation,

sinθ+cosθ=

2

…(1)

Taking square both sides,

sin

2

θ+cos

2

θ+2sinθcosθ=2

1+2sinθcosθ=2

2sinθcosθ=1

sinθcosθ=

2

1

……(2)

Now, divided by cosθ in equation 1st , we get

tanθ+1=

cosθ

2

….(3)

Again divided by sinθ in equation 1st, we get

1+cotθ=

sinθ

2

….(4)

Add equation 1st and 2nd , we get

tanθ+cotθ+2=

cosθ

2

+

sinθ

2

tanθ+cotθ=

2

.(

sinθcosθ

sinθ+cosθ

)−2 ……(5)

Now, from equation 1st ,2nd and 5th ,we get

tanθ+cotθ=

2

.

⎝

⎜

⎜

⎛

2

1

2

⎠

⎟

⎟

⎞

−2

tanθ+cotθ=2

Hence, this is the answer.