Question

If sec^4 theta -sec^2 theta = 10 + tan^4 theta + tan ^2 theta

then find the value of sin^2 theta

with full explanation​

Answer 1

(Instead of θ, I use A)

Answer:

Step-by-step explanation:

Given :

$sec^4A - sec^2A = 10 + tan^4A + tan^2A$

Then,

Find the value of :

$sin^2A$

Solution :

We know that,

$sec^2 A - tan^2 A = 1$

⇒ $sec^2 A = 1 + tan^2 A$, (note)

$sin^2A + cos^2A=1$

⇒ $sin^2A = 1 - cos^2A$  (note)

By equating the given equation,

$sec^4A - sec^2A = 10 + tan^4A + tan^2A$

⇒ $(sec^2A)^2 - sec^2A = 10 + tan^4A + tan^2A$

⇒ $sec^2A(sec^2A - 1) = 10 + tan^4A + tan^2A$

⇒ $sec^2A(tan^2A) = 10 + tan^4A + tan^2A$

⇒ $(1+tan^2A)(tan^2A) = 10 + tan^4A + tan^2A$

⇒ $tan^4A + tan^2A = 10 + tan^4A + tan^2A$

⇒ $tan^4A + tan^2A - tan^4A - tan^2A = 10$

⇒ $0 = 10$

Which is completely impossible (theoretically & logically)

So, The given equation is Incorrect,.

Hence, there is no existence of Sin²A , from the given Equation.

∴ Sin²A Does not exist,.

In case, if it was,

$sec^4A + sec^2A = 10 + tan^4A + tan^2A$

⇒ $(sec^2A)^2 + sec^2A = 10 + tan^4A + tan^2A$

⇒ $(sec^2A)(sec^2A + 1) = 10 + tan^4A + tan^2A$

⇒ $(1+tan^2A)(2+tan^2A) = 10 + tan^4A + tan^2A$

⇒ $2 + 3tan^2A + tan^4A = 10 + tan^4A + tan^2A$

⇒ $2 + 3tan^2A + tan^4A - tan^4A - tan^2A = 10$

⇒ $2 + 2tan^2A = 10$

⇒ $2tan^2A = 10 - 2$

⇒ $2tan^2A = 8$

⇒ $tan^2A = 4$

⇒ $1 + tan^2A = 4 + 1$

⇒ $sec^2A = 5$

⇒ $cos^2A = \frac{1}{5}$

⇒ $1 - cos^2A =1 - \frac{1}{5}$

⇒ $sin^2A = \frac{5 -1}{5} = \frac{4}{5}$

∴ $sin^2A = \frac{4}{5}$