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Introduction to Trigonometry

Short Answer Type

251.

If $secθ space equals space straight x plus fraction numerator 1 over denominator 4 straight x end fraction comma$ prove that $secθ plus tanθ space equals space 2 space straight x space or space fraction numerator 1 over denominator 2 straight x end fraction.$

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252.

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

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253. If x = y cos a. sin β, y = γ cos α.cos β, z = γ sin a. Show that x² + y² +z² = γ² = γ².

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254.

If $cosθ plus sinθ space equals space square root of 2 space cosθ comma$ show that $cosθ minus sinθ space equals space square root of 2 space sin space straight theta.$

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255. In the given Fig. ABCD is a rectangle with DC = 8 cm and BC = 6 cm. Line segment CM is drawn making an angle of 30° with DC, intersecting AD in M. Find the length of CM and DM.

Solution not provided.
Ans. $fraction numerator 16 square root of 3 over denominator 3 end fraction comma space space fraction numerator 8 square root of 3 over denominator 3 end fraction$

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256. In the given Fig. ABCD is a rectangle in which segments AP and AQ are drawn as shown. Find the length of (AP + AQ).

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257.

If $sin space straight theta plus space cos space straight theta space equals space square root of 2 space cos left parenthesis 90 degree minus straight theta right parenthesis.$ Determine $cot space straight theta.$

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258.

If $tanθ space equals space 1 space space and space sinβ space equals space fraction numerator 1 over denominator square root of 2 end fraction.$ Find the value of cos (θ + β), if θ and β are both acute angles.

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259.

If $fraction numerator cos space straight alpha over denominator cos space straight beta end fraction space equals space straight m space and space fraction numerator cos space straight beta over denominator sin space straight beta end fraction space equals space straight n comma$ show that $left parenthesis straight m squared plus straight n squared right parenthesis space cos squared straight beta space equals space straight n squared.$

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260. If x = a cos θ – b sin θ; y = a sin θ + b cos θ. Prove that x²+ y² = a² + b²

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