Find the length of the median from vertex A to side BC in a triangle with AB=8cm, AC=6cm and angle BAC=60°, using the cosine rule followed by the median length formula.
Prove that sin2A+sin2B−sin2C=4cosA cosB sinC when A+B+C=180°, using the sum-to-product formula twice along with the supplementary-angle identities.
Find the value of the product cos(π/4)cos(π/8)···cos(π/1024)sin(π/1024) using the repeated identity sinθcosθ=½sin2θ to telescope it down.
Simplify tan⁻¹(cosx/(1−sinx)) for x∈(−3π/2,π/2) using half-angle substitutions to reveal it equals tan(π/4+x/2), matching the principal range exactly.
Given cot x = −5/√11 with x in the second quadrant, evaluate (sin(11x/2))(sin6x−cos6x)+(cos(11x/2))(sin6x+cos6x) by collapsing it to sin(x/2)+cos(x/2).
A wheel makes 360 revolutions in one minute. Find how many radians it turns through in 1 second, converting revolutions per minute into radians per second.
Evaluate 2sin(12°)−sin(72°) by splitting 72° as 60°+12°, then recombining into a single sine of a difference angle using sin18°.
Given sinx+sin²x=1 for x in (0,π/2), evaluate a large trigonometric expression by first proving tanx=cosx, then spotting a binomial cube pattern.
Given sec²(tan⁻¹α)+cosec²(cot⁻¹β)=36 and α+β=8 with α<β, find α²+β by converting to a system of equations in α+β and αβ.