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

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In this video, we covered two questions

[1] If cos θ + sec θ = 1/2, then the value of cos3 θ +  sec3 θ is       

      (A) -1     (B) 0     (C) 1     (D)  -11/8   

[2] If cos θ + sec θ = 2, the value of cos6 θ +  sec6 θ is          SSC (10+2) 2012 

      (A) 4    (B) 8     (C) 1     (D)  2

                           

1 answer

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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.

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