when SECA = 3 where A is an acute angle.
: tan” 60° +4 sin® 45° + 3 sec? 60° +5¢
Evaluaicy cos ec30°+ sec 6...

Question

when SECA = 3 where A is an acute angle.
: tan” 60° +4 sin® 45° + 3 sec? 60° +5¢
Evaluaicy cos ec30°+ sec 60° — cot? 30°
1+tan’ A) (1-tanA)
gE Prove that 17 + col A) (1-cot A)*
PRIOR
MARK :
Multiple Choice Questions.
a 1
, If sin@ = then tan@ is equal to
) : b) 4s
a) —= —
45 9
If cosA = 2, then value of cotA.sinA 18:
5 HE
a) —== SURO V) :
If V25ing = 1, then cotfxcosech is eque

CramX · Accepted Answer

Step 1: Solution. {
  "solution": {
    "steps": [
      {
        "section": "Q3",
        "title": "Find cos A using sec A",
        "formula": "$\cos A = \frac{1}{\sec A} = \frac{3}{5}$",
        "explanation": "Since sec A is given as 5/3, cos A is the reciprocal."
      },
      {
        "section": "Q3",
        "title": "Find sin A using Pythagorean identity",
        "formula": "$\sin^2 A = 1 - \cos^2 A$",
        "explanation": "Use the identity to find sin A."
      },
      {
        "section": "Q3",
        "title": "Calculate sin A",
        "formula": "$\sin A = \sqrt{1 - \left(\frac{3}{5}ight)^2} = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5}$",
        "explanation": "Substitute cos A and simplify to get sin A."
      },
      {
        "section": "Q3",
        "title": "Final Answer",
        "formula": "$\sin A = \frac{4}{5}$",
        "explanation": "The value of sin A is 4/5 for the given sec A."
      },
      {
        "section": "Q4",
        "title": "Evaluate numerator",
        "formula": "$	an^2 60^\circ + 4\sin^2 45^\circ + 3\sec^2 60^\circ + 5$",
        "explanation": "Substitute values: tan 60° = √3, sin 45° = 1/√2, sec 60° = 2."
      },
      {
        "section": "Q4",
        "title": "Calculate each term in numerator",
        "formula": "$	an^2 60^\circ = 3$, $4\sin^2 45^\circ = 4 	imes \frac{1}{2} = 2$, $3\sec^2 60^\circ = 3 	imes 4 = 12$, $5$",
        "explanation": "Evaluate each trigonometric term."
      },
      {
        "section": "Q4",
        "title": "Sum numerator terms",
        "formula": "$3 + 2 + 12 + 5 = 22$",
        "explanation": "Add all terms for the numerator."
      },
      {
        "section": "Q4",
        "title": "Evaluate denominator",
        "formula": "$\csc 30^\circ + \sec 60^\circ - \cot^2 30^\circ$",
        "explanation": "Substitute values: csc 30° = 2, sec 60° = 2, cot 30° = √3."
      },
      {
        "section": "Q4",
        "title": "Calculate each term in denominator",
        "formula": "$\csc 30^\circ = 2$, $\sec 60^\circ = 2$, $\cot^2 30^\circ = 3$",
        "explanation": "Evaluate each trigonometric term."
      },
      {
        "section": "Q4",
        "title": "Sum denominator terms",
        "formula": "$2 + 2 - 3 = 1$",
        "explanation": "Add and subtract terms for the denominator."
      },
      {
        "section": "Q4",
        "title": "Final Answer",
        "formula": "$\frac{22}{1} = 22$",
        "explanation": "The evaluated value is 22."
      },
      {
        "section": "Q5",
        "title": "Express tan²A and cot²A in terms of tanA",
        "formula": "$\cot^2 A = \frac{1}{	an^2 A}$",
        "explanation": "Rewrite cot²A using tanA."
      },
      {
        "section": "Q5",
        "title": "Rewrite LHS",
        "formula": "$\frac{1 + 	an^2 A}{1 + \cot^2 A} = \frac{1 + 	an^2 A}{1 + \frac{1}{	an^2 A}}$",
        "explanation": "Substitute cot²A."
      },
      {
        "section": "Q5",
        "title": "Simplify denominator",
        "formula": "$1 + \frac{1}{	an^2 A} = \frac{	an^2 A + 1}{	an^2 A}$",
        "explanation": "Combine terms in the denominator."
      },
      {
        "section": "Q5",
        "title": "Simplify overall expression",
        "formula": "$\frac{1 + 	an^2 A}{\frac{	an^2 A + 1}{	an^2 A}} = (1 + 	an^2 A) 	imes \frac{	an^2 A}{	an^2 A + 1} = 	an^2 A$",
        "explanation": "Multiply numerator by reciprocal of denominator."
      },
      {
        "section": "Q5",
        "title": "Rewrite RHS",
        "formula": "$\frac{(1 - 	an A)^2}{(1 - \cot A)^2}$",
        "explanation": "Express RHS in terms of tanA and cotA."
      },
      {
        "section": "Q5",
        "title": "Express cotA in terms of tanA",
        "formula": "$\cot A = \frac{1}{	an A}$",
        "explanation": "Substitute cotA."
      },
      {
        "section": "Q5",
        "title": "Rewrite denominator of RHS",
        "formula": "$(1 - \cot A)^2 = \left(1 - \frac{1}{	an A}ight)^2 = \left(\frac{	an A - 1}{	an A}ight)^2$",
        "explanation": "Express denominator in terms of tanA."
      },
      {
        "section": "Q5",
        "title": "Rewrite numerator of RHS",
        "formula": "$(1 - 	an A)^2$",
        "explanation": "Numerator remains as is."
      },
      {
        "section": "Q5",
        "title": "Combine numerator and denominator",
        "formula": "$\frac{(1 - 	an A)^2}{\left(\frac{	an A - 1}{	an A}ight)^2} = (1 - 	an A)^2 	imes \frac{	an^2 A}{(	an A - 1)^2}$",
        "explanation": "Multiply numerator by reciprocal of denominator."
      },
      {
        "section": "Q5",
        "title": "Simplify RHS",
        "formula": "$	an^2 A$",
        "explanation": "The terms (1 - tanA)^2 and (tanA - 1)^2 cancel out, leaving tan²A."
      },
      {
        "section": "Q5",
        "title": "Final Answer",
        "formula": "$\frac{1 + 	an^2 A}{1 + \cot^2 A} = \frac{(1 - 	an A)^2}{(1 - \cot A)^2}$",
        "explanation": "Both sides simplify to tan²A, hence proved."
      },
      {
        "section": "MCQ1",
        "title": "Given sin θ = 1/9, find tan θ",
        "formula": "$\sin 	heta = \frac{1}{9}$",
        "explanation": "Let the opposite side be 1, hypotenuse be 9."
      },
      {
        "section": "MCQ1",
        "title": "Find adjacent side using Pythagoras",
        "formula": "$	ext{adjacent} = \sqrt{9^2 - 1^2} = \sqrt{81 - 1} = \sqrt{80} = 4\sqrt{5}$",
        "explanation": "Calculate the adjacent side."
      },
      {
        "section": "MCQ1",
        "title": "Calculate tan θ",
        "formula": "$	an 	heta = \frac{1}{4\sqrt{5}}$",
        "explanation": "Tan θ is opposite over adjacent."
      },
      {
        "section": "MCQ1",
        "title": "Final Answer",
        "formula": "$	an 	heta = \frac{1}{4\sqrt{5}}$",
        "explanation": "Correct option is (a)."
      },
      {
        "section": "MCQ2",
        "title": "Given cos A = 5/8, find sin A",
        "formula": "$\sin^2 A = 1 - \cos^2 A$",
        "explanation": "Use Pythagorean identity."
      },
      {
        "section": "MCQ2",
        "title": "Calculate sin A",
        "formula": "$\sin A = \sqrt{1 - \left(\frac{5}{8}ight)^2} = \sqrt{1 - \frac{25}{64}} = \sqrt{\frac{39}{64}} = \frac{\sqrt{39}}{8}$",
        "explanation": "Find sin A."
      },
      {
        "section": "MCQ2",
        "title": "Find cot A",
        "formula": "$\cot A = \frac{\cos A}{\sin A} = \frac{5/8}{\sqrt{39}/8} = \frac{5}{\sqrt{39}}$",
        "explanation": "Cot A is cos A over sin A."
      },
      {
        "section": "MCQ2",
        "title": "Calculate cotA.sinA",
        "formula": "$\cot A \cdot \sin A = \frac{5}{\sqrt{39}} \cdot \frac{\sqrt{39}}{8} = \frac{5}{8}$",
        "explanation": "Multiply cot A and sin A."
      },
      {
        "section": "MCQ2",
        "title": "Final Answer",
        "formula": "$\cot A \cdot \sin A = \frac{5}{8}$",
        "explanation": "Correct option is (b)."
      },
      {
        "section": "MCQ3",
        "title": "Given $\sqrt{2} \sin 	heta = 1$, find sin θ",
        "formula": "$\sin 	heta = \frac{1}{\sqrt{2}}$",
        "explanation": "Divide both sides by √2."
      },
      {
        "section": "MCQ3",
        "title": "Find cosec θ",
        "formula": "$\csc 	heta = \frac{1}{\sin 	heta} = \sqrt{2}$",
        "explanation": "Cosec θ is reciprocal of sin θ."
      },
      {
        "section": "MCQ3",
        "title": "Find cot θ",
        "formula": "$\cot 	heta = \frac{\cos 	heta}{\sin 	heta}$",
        "explanation": "Use definition of cot θ."
      },
      {
        "section": "MCQ3",
        "title": "Find cos θ using Pythagoras",
        "formula": "$\cos^2 	heta = 1 - \left(\frac{1}{\sqrt{2}}ight)^2 = 1 - \frac{1}{2} = \frac{1}{2}$, $\cos 	heta = \frac{1}{\sqrt{2}}$",
        "explanation": "Calculate cos θ."
      },
      {
        "section": "MCQ3",
        "title": "Calculate cot θ",
        "formula": "$\cot 	heta = \frac{1/\sqrt{2}}{1/\sqrt{2}} = 1$",
        "explanation": "Cot θ is cos θ over sin θ."
      },
      {
        "section": "MCQ3",
        "title": "Calculate cot θ × cosec θ",
        "formula": "$1 	imes \sqrt{2} = \sqrt{2}$",
        "explanation": "Multiply cot θ and cosec θ."
      },
      {
        "section": "MCQ3",
        "title": "Final Answer",
        "formula": "$\sqrt{2}$",
        "explanation": "The value is √2."
      }
    ]
  }
}

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