Question

evaluate 3 cot²60° +sec² 45°​

Answer 1

Question :

Evaluate 3cot²60° +sec² 45°

$\blue{\huge{\boxed{\sf{Solution:}}}}$

Let's look at some values

$\begin{array}{| c | c | c | c | c | c |}\cline{1-6} \bf Angles & \bf 0^{o} & \bf 30^{o} & \bf 45^{o} & \bf 60^{o} & \bf 90^{o} \\\cline{1-6} \tt Sin \theta & 0 & \dfrac{1}{2} & \dfrac{1}{\sqrt{2}}& \dfrac{\sqrt{3}}{2}& 1\\\cline{1-6} \tt cos \theta & 1 & \dfrac{\sqrt{3}}{2} &\dfrac{1}{\sqrt{2}}&\dfrac{1}{2}&0\\\cline{1-6} \tt tan \theta & 0 & \dfrac{1}{\sqrt{3}} & 1& \sqrt{3} & \infty \\\cline{1-6} \tt cosec \theta & \infty & 2 & \sqrt{2} & \dfrac{2}{\sqrt{3}} &1\\\cline{1-6} \tt sec \theta & 1 & \dfrac{2}{\sqrt{3}} & \sqrt{2} & 2 & \infty \\\cline{1-6} \tt cot \theta & \infty & \sqrt{3} &1& \dfrac{1}{\sqrt{3}} &0\\\cline{1-6}\end{array}$

Here, '$\red{\infty}$' stands for

undefined(not defined) values.

$\rule{200}2$

We have to find the value of

$\sf3\cot^260\degree+\sec^245\degree$

$\sf=3(\cot60\degree)^2+(\sec45\degree)^2$

Now put the values of cot60° and sec45°

$\sf=3\times(\dfrac{1}{\sqrt{3}})^2+(\sqrt{2})^2$

$\sf=3\times\dfrac{1}{3}+2$

$\sf=1+2$

$\sf=3$

Answer 2

Hey there!

3 cot²60° +sec² 45°​

3×(1/√3)²+(√2)²

3×(1/3)+2

1+2

3 is the answer.

HERE IS YOUR EXPLANATION

As we know that cot 60° = 1/√3

So, 3×cos²60 = 3×(1/√3)²

Now when we square (1/√3) we get (1/3)

So, 3×(1/3) will give the value, 1.

For the next part,

sec 45°​ = √2 (its a fixed value from the table)

so, sec² 45°​= (√2)²

which is equal to 2.

so as per the question,

3 cot²60° +sec² 45°​

our answer will be 1+2= 3