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Introduction to Trigonometry

Short Answer Type

281. Show that 2 sec² θ – sec⁴ θ – 2 cosec² θ + cosec⁴θ = cot⁴ θ – tan⁴ θ.

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282. Show that:

fraction numerator cosA over denominator 1 minus tanA end fraction plus fraction numerator sinA over denominator 1 minus cotA end fraction equals sinA plus cosA.

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283.

Prove that $fraction numerator sin space straight theta over denominator 1 plus cosθ end fraction plus fraction numerator 1 plus cosθ over denominator sinθ end fraction space equals space 2 space cosecθ.$

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284. If 3 tan θ = 4. Find the value of

fraction numerator 2 space sin space straight theta space plus space 3 space cos space straight theta over denominator 2 space sin space straight theta space minus space 3 space cos space straight theta end fraction.

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285. Prove that: (sin θ + sec θ)² + (cos θ + cosec θ)² = (1 + sec θ. cosec θ)².

Solution not provided.

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286. If 4 tan θ = 3, find the value of

fraction numerator 4 space sin space straight theta space minus space 2 space cosθ over denominator 4 space sinθ space plus space 3 space cosθ end fraction.

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287. Prove that:

fraction numerator cos squared straight theta over denominator 1 minus tanθ end fraction plus fraction numerator sin cubed straight theta over denominator sin space straight theta space minus space cos space straight theta end fraction space equals space 1 plus sinθ. space cosθ.

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288. Prove that:

fraction numerator cos squared 20 degree plus cos squared 70 degree over denominator sin squared 20 degree space plus space sin squared 70 degree end fraction plus sin squared 64 degree plus cos 64 degree. space sin space 26 degree space equals space 2.

Find the value of sin² θ – cos² θ.

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289.

If $square root of 3 space tan space straight theta space equals space 3 space sin space straight theta.$ Find the value of sin² θ – cos² θ.

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290.

Prove that:
$fraction numerator tanθ plus secθ minus 1 over denominator tanθ minus secθ plus 1 end fraction equals fraction numerator 1 plus sinθ over denominator cosθ end fraction.$