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Mathematics : Introduction to Trigonometry

Short Answer Type

241. Evaluate:
sin 63°.cos 27° + cos 63°. sin 27°

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242. Evaluate:
sin 48° . sec 42° + cos 48° . cosec 42°.

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243.

If $cos squared straight alpha space minus space sin 2 straight alpha space equals space tan squared straight beta comma$ then prove that $cos space straight beta space equals space fraction numerator 1 over denominator square root of 2 cosα end fraction.$

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244. If cosec θ - sin θ = m³ and sec θ - cos θ = n³ then prove that m⁴n² + m²n⁴ = 1.

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245. Simplify:

fraction numerator sin space 73 degree over denominator cos space 17 degree end fraction plus 3 fraction numerator sec space 31 degree over denominator cosec space 59 degree end fraction minus 2 fraction numerator left parenthesis sin squared 41 degree plus sin squared 49 degree right parenthesis over denominator sec squared 20 degree minus cot squared 70 degree plus 3 tan squared 30 degree end fraction.

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246. If cosec θ - sin θ = a, sec θ - cos θ = b. Prove that a² b² (a² + b² + 3) = 1.

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247. Show that (sec θ - cosec θ) (1+ tan θ + cot θ) = sec a².tan θ - cosec θ.cot θ.

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248. If cosec θ - sin θ = a, sec θ - cos θ = b. Prove that a² b² (a² + b² + 3) = 1.

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249. If

secθ space equals space straight x plus fraction numerator 1 over denominator 4 straight x end fraction comma

Prove that

secθ plus tanθ space equals space 2 straight x space or space fraction numerator 1 over denominator 2 straight x end fraction.

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250.

Show that $fraction numerator cosec space straight theta space plus space cotθ over denominator cosecθ space minus space cotθ end fraction space equals space 1 plus 2 cot squared straight theta plus 2 cosecθ. space cotθ$

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