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Prove that (cos x+cos y)²+(sin x−sin y)²=4cos²((x+y)/2).
Expand both squares and combine the cross terms.
Differentiate sin(x+a).
Name the inner functions, differentiate each factor, and then simplify.
Substitution into the original expression verifies the stated domain and final value.
A hyperbola has foci (±5,0) and transverse-axis length 8. Find its equation.
Translate every condition into an equation and retain exact values throughout the hyperbola calculation.
The last line satisfies the original conditions and supplies every requested part.
Find both lines through (3,2) making 45° with x−2y=3.
Translate every condition into an equation and retain exact values throughout the straight-line calculation.
The last line satisfies the original conditions and supplies every requested part.
Find the n-term sums of (i) 5+55+555+⋯(ii) 0.6+0.66+0.666+⋯ .
Translate the conditions into a sequence equation, then retain every admissible real solution.
The final value satisfies the original recurrence, sum, product or financial condition.
Complete every requested part of miscellaneous exercise, question 4: (cos x – cos y)2 + (sin x – sin y)2 = 4 sin2 2
Proceed from the given expression and keep every transformation explicit.
The last line satisfies the required domain and establishes the result.
If \(0.561\,\mathrm{g}\) of KOH is dissolved in
KOH is a strong base. Convert mass to moles and divide by 0.200 L. Use Kw to find [H⁺] and pH=14−pOH.
An electron microscope uses electrons moving at 1.6 × 10⁶ \(\mathrm{m\,s}\)⁻¹. Calculate their de Broglie wavelength.
Apply λ=h/(mₑv).
Differentiate cosec x cot x.
Name the inner functions, differentiate each factor, and then simplify.
Substitution into the original expression verifies the stated domain and final value.
A hyperbola has foci (0,±13) and conjugate-axis length 24. Find its equation.
Translate every condition into an equation and retain exact values throughout the hyperbola calculation.
The last line satisfies the original conditions and supplies every requested part.