Question

If `tan^2theta = 1 - a^2`, then `sectheta + tan^3thetacsctheta = (2 - a^2)^n`. What is the value of `n`?

Answer 1

`1.5`

Explanation:

`a^2 = 1 - tan^2theta`

Substitute:

`sectheta + tan^3thetacsctheta = (2 - (1 - tan^2theta))^n`

Apply the following identities:

`•secbeta = 1/cosbeta`
`•tan beta = sinbeta/cosbeta`
`•cscbeta = 1/sinbeta`

`1/costheta + sin^3theta/cos^3theta xx 1/sintheta = (2 - (1 - tan^2theta))^n`

`1/costheta + sin^2theta/cos^3theta = (1 + tan^2theta)^n`

Put the left hand side on a common denominator and apply the identity `1 + tan^2beta = sec^2beta`.

`(cos^2theta + sin^2theta)/cos^3theta = (sec^2theta)^n`

Use the identity `sin^2beta + cos^2beta = 1`.

`1/cos^3theta = (sec^2theta)^n`

`sec^3theta = (sec^2theta)^n`

We can call `sectheta = x`.

`x^3 = (x^2)^n`

`x^3 = x^(2n)`

`3 = 2n`

`n = 1.5`

Hopefully this helps!

Answer 2

`3/2` and `theta in Q_1`.

Explanation:

Let `t = tan theta`. Then. `t^2=1-a^2>=0 to |a|<=1`

Now,

`sec theta+tan^3theta csc theta`

`=+-sqrt(1+t^2)+t^3(+-sqrt(I+1/t^2))`

`=+-sqrt(1+t^2)(1+-t^2)`

`=+-(1+t^2)^(3/2)`, for + sign in the second factor ( `csc theta > 0`)

`=+-(2-a^2)^(3/2 )`, `-` sign is taken when `sec theta < 0`

`=(2-a^2)^n to sec theta > 0 and`

`n=3/2` and csc theta >0 to theta in Q_1#.

I am sorry that I have involved `+-`. But, this is Mathematics.

I have obtained an enforcing result `theta in Q_1`.

Answer 3

`n = 3/2`

Explanation:

`1/costheta+sin^3theta/cos^3theta 1/sintheta=1/costheta(1+tan^2theta)=(2-a^2)^n`

then

`costheta=(2-a^2)/(2-a^2)^n=(2-a^2)^(-(n-1))`

squaring

`cos^2theta=(2-a^2)^(-2(n-1))`

but from `tan^2theta=(1-a^2)->cos^2theta=(2-a^2)^(-1)`

arriving at

`(2-a^2)^(-2(n-1))=(2-a^2)^(-1)` so

`2(n-1)=1->n=3/2`

Answer 4

Given

`tan^2theta=1-a^2`

`1+tan^2theta=1+1-a^2`

`sec^2theta=2-a^2`

Now
`sectheta+tan^3thetacsctheta=(2-a^2)^n`

`=>sectheta+tan^3thetaxx1/sintheta=(2-a^2)^n`

`=>sectheta(1+tan^3thetaxxcostheta/sintheta)=(2-a^2)^n`

`=>sectheta(1+tan^3thetaxx1/tantheta)=(sec^2theta)^n`

`=>sectheta(1+tan^2theta)=(sec^2theta)^n`

`=>secthetaxxsec^2theta=sec^(2n)theta`

`=>sec^3theta=sec^(2n)theta`

`=>2n=3`

`:.n=3/2`