# If cos θ + sec θ = 2, then the value of cos^6 θ + sec^6 θ is

In this video, we will solve 2 questions from Trigonometry.

We will understand the basic concepts.

And, also learn the short trick to solve question faster.

[1] So, let us take our first question.

If cos θ + sec θ is equal to one half, then the value of cos^3 θ plus sec^3 θ is.

First, we will understand the concept.

Let's take cos θ equal to x then sec θ equal to 1 over x.

Cos θ + sec θ equal to one half given.

We can write. x plus one over x equal to one half.

We need to find the value of cos^3 θ + sec^3 θ.

We can write. x^3 + 1/ x^3.

Next, cube both sides.

We have, (x + 1/x)^3 = (1/2)^3.

We can expand this way.

X^3 + 1/x^3 + 3 (x) (1/x) (x + 1/x) = 1/8.

Next, we can cancel x. Now, we are left with this. (Watch Video)

We can transfer this to the right side. (Watch Video)

We can write, x^3 + 1/x^3 = (1/8) - 3 (x + 1/x).

Remember this identity for faster calculation. (Watch Video)

Now, replace x + 1/x by one half.

We have x^3 + 1/x^3 = (1/8) - (3/2).

Here LCM is 8. And you can easily subtract fractions this way. (Watch Video)

Finally, we got cos^3 θ + sec^3 θ = - 11/8.

Now, you understood the concept.

You can solve this question faster with the short trick.

Cos^3 θ + sec^3 θ = (cos θ + sec θ)^3 - 3 ( cos θ + sec θ).

Now, the cos θ plus sec θ = 1/2.

So, we have (1/2)^3 - 3 (1/2).

Next, we have (1/8) - (3/2).

Subtract the fractions and you have the final answer.

Cos^3 θ + sec^3 θ = -11/8.

So, the correct answer is Option (D) - 11/8.

[2] Now, let us solve the second question.

If cos θ + sec θ = 2, the value of cos^6 θ + sec^6 θ is.

Now, let's solve this.

The cos θ + sec θ = 2.

We can write sec θ = 1/ cos θ =1.

Take LCM.

And, we can write.

( cos^2 θ + 1) /cos θ =2.

Next, cross multiply.

After, cross multiplication, we have cos^2 θ + 1 = 2 (cos θ). Transfer this to the left side. (watch video)

cos^2 θ + 1 - 2 cos θ =0. Next, you can apply this algebraic identity. (watch video)

And, write (cos θ -1)^2 = zero.

Then, the cos θ - 1 =0.

It gives us, the cos θ =1. And. the sec θ =1.

So, we can write 1^6 + 1^6 =2.

So, the correct answer is the option (D) 2.