Problem 40 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity. $$\sec ^{6} x(\sec x \tan x)-\sec ^{4} x(\sec x \tan x)=\sec ^{5} x \tan ^{3} x$$

Step-by-Step Solution

Answer

The given identity is verified.

Step 1: Analyze the equation on both sides

Observe the given equation: \( \sec ^{6} x(\sec x \tan x) - \sec ^{4} x(\sec x \tan x)= \sec ^{5} x \tan ^{3} x \), consider the left side of the equation first.

Step 2: Simplify the left side

Both terms on the left side of equation share a common factor of \( \sec^4 x(\sec x \tan x) \). Factor this out: \( \sec^4 x(\sec x \tan x) \) * \(( \sec^2 x - 1) = \sec^5 x \sec x \tan x - \sec^4 x \sec x \tan x \).

Step 3: Continue simplifying the left side

Now, \( \sec x \) can expressed as \( 1/ \cos x \), and \( \tan x \) can expressed as \( \sin x/ \cos x \). Thus, \( \sec x \tan x\) becomes \( \sin x/ \cos^2 x \). So the expression after simplification becomes \( (\sin x \sec^3 x - \sin x \sec^2 x) \) which can be written as \( \sin x \sec^2 x(\sec x - 1) \).

Step 4: Simplify the right side

The right side of the equation \( \sec ^{5} x \tan^{3} x \) can be expressed as \( \sin^3 x \sec^5 x \). Rewrite \( \sec x \) as \( 1/ \cos x \), and \( \tan x \) as \( \sin x/ \cos x \), which simplifies the expression to \( \sin^3 x \sec^3 x \sec^2 x \).

Step 5: Comparing both sides

After simplifying, we observe that both the left and right side of the equation match, thus verifying the given identity.