Question

Prove the give identity: sec^(4)A+tan^(4)A=1+2tan^(2)A*sec^(2)A

Answer 1

Answer:

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Step-by-step explanation:

To prove or verify that the given trigonometric equation is an identity, we'll need to utilize one or more of the basic trigonometric identities and also utilize algebraic manipulations as follows:

(1.) sec^4 x + tan^4 x = 1 + 2sec^2 x tan^2 x (Given)

(2.) (sec^2 x)^2 + (tan^2 x)^2 = 1 + 2sec^2 x tan^2x

Since one of our basic trigonometric identities is: 1 + tan^2 x = sec^2 x, we can substitute into equation (2.) as follows:

(3.) (1 + tan^2 x)^2 + (tan^2 x)^2 = 1 + 2(1 + tan^2 x) tan^2x)

(4.) 1 + 2tan^2 x + tan^4 x + (tan^2 x)^2 = 1 + 2(tan^2 x + tan^4 x)

(5.) 1 + 2tan^2 x + tan^4 x + tan^4 x = 1 + 2tan^2 x + 2tan^4 x

Collecting like terms on the left, we get:

(6.) 1 + 2tan^2 x + 2tan^4 x = 1 + 2tan^2 x + 2tan^4 x

Looking at equation (6.), we now see that given trigonometric equation, sec^4 x + tan^4 x = 1 + 2sec^2 x tan^2 x, has been proven or verified to be an identity that is therefore true for all permissible values of angle x.