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Challenge 344: Some Length-y Trigonometry

(i) Use a calculator to check that sin 15o + √2 = √3 cos 15o

(ii) Triangle PQR has a right-angle at R, and the angle at Q is 60o.

Triangle PQS is a right-angled isosceles triangle, with the right-angle at S.

S is the same side of PQ as R, and QS crosses PR at the point T.

Using a diagram of these two triangles, or otherwise, prove that sin 15o + √2 = √3 cos 15o without using a calculator or other technology.