Question

if tan theta + cot theta = 2 then tan ^2020 theta + cot ^ 2020 theta is​

Answer 1

Step-by-step explanation:

Answer

Correct option is

A

2

tanθ+cotθ=2

Squaring both sides;

(tanθ+cotθ)

2

=2

2

⇒tan

2

θ+cot

2

θ+2tanθcotθ=4

⇒tan

2

θ+cot

2

θ+2=4

⇒tan

2

θ+cot

2

θ=2

Answer 2

Given:

Tan θ + Cot θ = 2

To find:

The value of Tan²⁰²⁰ θ + Cot²⁰²⁰ θ.

Solution:

It is given that,

$tan \theta +cot \theta =2$

we know that the reciprocal of $Cot\theta$ is $tan\theta$ i.e.,

$Cot\theta=\frac{1}{tan\theta}$

Substituting in the given expression:

$Tan\theta +\frac{1}{Tan\theta}=2$

Taking the LCM as $Tan\theta$

$\frac{Tan^{2} \theta+1}{Tan\theta} =2$

On cross-multiplying,

$Tan^{2} \theta +1=2Tan\theta$

Transposing $2Tan\theta$ from RHS to LHS

$Tan^{2} \theta-2Tan\theta +1=0$

This is in the form of identity,

$(a-b)^{2} =a^{2} -2ab+b^{2}$

where, $a=Tan\theta, b=1$

$(Tan\theta -1)^{2} =0$

$Tan\theta -1=0$

$Tan\theta=1$

Now, $Cot\theta=\frac{1}{tan\theta}$

$Cot\theta=\frac{1}{1} =1$

∴ $Tan\theta=1, Cot\theta=1$

Then,

$Tan^{2020}\theta+Cot^{2020}\theta$

$=1^{2020}+1^{2020}$

Any exponent of 1 as the base is 1 itself.

Hence, $=1^{2020}+1^{2020}=1+1=2$

Thus, $Tan^{2020}\theta+Cot^{2020}\theta=2$

If tan θ + cot θ = 2 then, tan²⁰²⁰ θ + cot²⁰²⁰ θ =​ 2