Prove that =cos5x−cosxsin5x−2sin3x+sinx=tanx
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Step-by-Step Solution
Step 1: Rearrange the numerator
We rearrange the terms in the numerator to group sin5x and sinx together. This allows us to apply the sum-to-product formula for sine.
Step 2: Apply sum-to-product formulas
We will use the sum-to-product formula for sine in the numerator and the difference-to-product formula for cosine in the denominator. These identities help simplify trigonometric expressions.
Step 3: Simplify the numerator
Applying the sum-to-product formula to (sin5x+sinx) gives 2sin3xcos2x. Then, we factor out 2sin3x from the expression.
Step 4: Simplify the denominator
Applying the difference-to-product formula to (cos5x−cosx) gives −2sin3xsin2x.
Step 5: Substitute and simplify
Substitute the simplified numerator and denominator back into the original expression. We can cancel out 2sin3x from both the numerator and the denominator.
Step 6: Use double angle identities
We use the double angle identities for cos2x and sin2x to further simplify the expression. Specifically, cos2x−1=−2sin2x.
Step 7: Final simplification
Substitute the double angle identities into the expression. Cancel out −2sinx from the numerator and denominator. The remaining term cosxsinx is equal to tanx.