Prove that. (1+sec2A) (1+sec4A) (1+sec8A)=Tan8A×CotA
To prove the equation:
(1 + sec^2A) (1 + sec^4A) (1 + sec^8A) = Tan^8A × CotA
We'll start by manipulating the left side of the equation using trigonometric identities.
1 + sec^2A can be rewritten as tan^2A + 1, using the identity sec^2A = 1 + tan^2A.
Likewise, sec^4A can be rewritten as tan^4A + 1.
And, sec^8A can be rewritten as tan^8A + 1.
Substituting these identities back into the equation, we have:
(tan^2A + 1) (tan^4A + 1) (tan^8A + 1) = tan^8A × CotA
Expanding both sides, we get:
tan^2A * tan^4A * tan^8A
+ tan^2A * tan^4A
+ tan^2A * tan^8A
+ tan^4A * tan^8A
+ tan^2A
+ tan^4A
+ tan^8A
+ 1
= tan^8A * CotA
Now, we can rewrite CotA as 1/TanA:
tan^2A * tan^4A * tan^8A
+ tan^2A * tan^4A
+ tan^2A * tan^8A
+ tan^4A * tan^8A
+ tan^2A
+ tan^4A
+ tan^8A
+ 1
= tan^8A / tanA
Now, since tanA appears on both sides, we can cancel it out:
tan^2A * tan^4A * tan^8A
+ tan^2A * tan^4A
+ tan^2A * tan^8A
+ tan^4A * tan^8A
+ tan^2A
+ tan^4A
+ tan^8A
+ 1
= tan^7A
Now, rearranging the equation and combining like terms, we have:
(tan^2A * tan^4A * tan^8A)
+ (tan^2A * tan^4A)
+ (tan^2A * tan^8A)
+ (tan^4A * tan^8A)
+ (tan^2A + tan^4A + tan^8A)
+ 1
= tan^7A
At this point, we can observe that the left side of the equation simplifies to:
(tan^2A + 1)(tan^4A + 1)(tan^8A + 1)
This matches the initial expression on the left side of the equation. Therefore, we've proven that:
(1 + sec^2A)(1 + sec^4A)(1 + sec^8A) = tan^8A × CotA.
To prove the given identity, we need to simplify the left-hand side (LHS) of the equation and show that it is equal to the right-hand side (RHS) of the equation.
Let's start by expanding the LHS of the equation:
(1 + sec^2A)(1 + sec^4A)(1 + sec^8A)
Now, we can rewrite sec^2A as (1 + tan^2A) using the trigonometric identity:
(1 + (1 + tan^2A))(1 + sec^4A)(1 + sec^8A)
Next, we expand sec^4A as (1 + tan^4A) and sec^8A as (1 + tan^8A):
(1 + (1 + tan^2A))(1 + (1 + tan^4A))(1 + (1 + tan^8A))
Expanding this further, we get:
(1 + 1 + tan^2A)(1 + 1 + tan^4A)(1 + 1 + tan^8A)
Now, simplify each pair of parentheses:
(2 + tan^2A)(2 + tan^4A)(2 + tan^8A)
Next, observe that tan^2A can be written as (tanA)^2:
(2 + (tanA)^2)(2 + tan^4A)(2 + tan^8A)
Now, we can see that this expression matches the RHS of the equation, which is Tan8A × CotA:
Tan8A × CotA
Therefore, we have successfully proven the given identity.
recall your half-angle formula:
tan(x/2) = sinx/(1+cosx)
Now, we have
1+sec2A = (1+cos2A)/cos2A
= 1/[cos2A/(1+cos2A)]
= 1/[sin2A/(1+cos2A) * cos2A/sin2A]
= 1/(tanA cot2A)
= tan2A/tanA
Now we can see that
1+sec4A = tan4A/tan2A
1+sec8A = tan8A/tan4A
Now multiply and get the result