Prove that sin7x−sin5xcos7x+cos5x=cotx
Get the complete, step-by-step math solution for: "Prove that (cos 7 x+cos 5 x)/(sin 7 x-sin 5 x)=cot x". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Apply Sum-to-Product Formulas
To simplify the given expression, we will use the sum-to-product trigonometric identities. The numerator involves the sum of two cosines, and the denominator involves the difference of two sines. We apply the respective formulas to transform these sums and differences into products.
Step 2: Substitute into the Expression
Now, we substitute A=7x and B=5x into the sum-to-product formulas. This transforms the original fraction into a new form where the numerator and denominator are products of trigonometric functions.
Step 3: Simplify the Angles
We perform the arithmetic operations within the arguments of the trigonometric functions. This simplifies the angles to 6x and x, making the expression easier to work with.
Step 4: Cancel Common Terms
We observe that 2cos(6x) is a common factor in both the numerator and the denominator. Assuming cos(6x)=0, we can cancel this common term, which significantly simplifies the expression.
Step 5: Apply Cotangent Identity
Finally, we use the fundamental trigonometric identity that states sinθcosθ=cotθ. Applying this identity to our simplified expression, we arrive at the desired result.