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Mathematics : Introduction to Trigonometry

Short Answer Type

291.

If $tanθ space equals space 4 over 5 comma$ find the value of $fraction numerator 2 space sinθ minus 3 cosθ over denominator 4 sinθ minus 9 cosθ end fraction.$

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292. Without using trigonometric table, evaluate the following : tan 5° . tan 25° . tan 30° . tan 65° . tan 85°.

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293. Prove the identity:

fraction numerator sinθ plus 1 minus cosθ over denominator cosθ minus 1 plus sinθ end fraction equals fraction numerator 1 plus sinθ over denominator cosθ end fraction.

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294. Prove the identity: (1 – sin θ + cos θ)² = 2(1 + cos θ) (1 – sin θ).

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295. Evaluate:

4 left parenthesis sin to the power of 4 30 degree plus cos to the power of 4 60 degree right parenthesis minus 2 over 3 left parenthesis sin squared 60 degree minus cos squared 45 degree right parenthesis plus 1 half tan squared 60 degree

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Long Answer Type

296. Prove the identity:

square root of fraction numerator 1 plus cosθ over denominator 1 minus cosθ end fraction end root plus square root of fraction numerator 1 minus cosθ over denominator 1 plus cosθ end fraction end root space equals space 2 space cosecθ.

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297. Prove the identity:

square root of fraction numerator 1 plus sinθ over denominator 1 minus sinθ end fraction end root plus square root of fraction numerator 1 minus sinθ over denominator 1 plus sinθ end fraction end root space equals space 2 space secθ.

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298. Evaluate:

1 fourth open square brackets cot to the power of 4 30 degree minus cosec to the power of 4 60 degree close square brackets plus 3 over 2 left square bracket sec squared 45 degree minus tan squared 30 degree minus 5 cos squared 60 degree right square bracket.

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Short Answer Type

299. Show that:

fraction numerator tanθ plus secθ minus 1 over denominator tanθ minus secθ plus 1 end fraction equals fraction numerator 1 plus sinθ over denominator cos space straight theta end fraction.

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300. Without using trigonometric table, evaluate:
sin (50° + θ) – cos (40° – θ) + tan 1°. tan 10° tan 20°. tan 70°. tan 80°. tan 89°.

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