Question

If cos theta =5/13 find sin^2theta-cosec^2theta+cot^2theta

Answer 1

Answer:

your answer attached in the photo

Answer 2

Question :

$\sf{If\:cos\:\theta=\frac{5}{13},\:find\:sin^2\:\theta-cosec^2\:\theta+cot^2\:\theta.}$

Solution :

$\sf{cos\:\theta=\frac{5}{13}}$

$\implies\sf{cos^2\:\theta=\frac{25}{169}}$

We know,

${\boxed{\bold{sin^2\:\theta=1-cos^2\:\theta}}}$

So,

$\implies\sf{sin^2\:\theta=1-(\frac{5}{13})^2}$

$\implies\sf{sin^2\:\theta=1-\frac{25}{169}}$

$\implies\sf{sin^2\:\theta=\frac{144}{169}}$

★${\boxed{\bold{sin^2\:\theta=\frac{144}{169}}}}$

We know,

${\boxed{\bold{cosec^2\:\theta=\dfrac{1}{sin^2\:\theta}}}}$

So,

$\implies\sf{cosec^2\:\theta=\frac{169}{144}}$

We know,

${\boxed{\bold{cot^2\:\theta=\dfrac{cos^2\:\theta}{sin^2\:\theta}}}}$

So,

$\implies\sf{cot^2\:\theta=\frac{25}{169}\times\:\frac{169}{144}}$

$\implies\sf{cot^2\:\theta=\frac{25}{144}}$

✪ Now find the value of sin²∅ - cosec²∅ + cot²∅ ✪

$\sf{sin^2\:\theta-cosec^2\:\theta+cot^2\:\theta}$

$\implies\sf{\frac{144}{169}-\frac{169}{144}+\frac{25}{144}}$

$\implies\sf{\frac{144}{169}-(\frac{169}{144}-\frac{25}{144})}$

$\implies\sf{\frac{144}{169}-\frac{144}{144}}$

$\implies\sf{\frac{144}{169}-1}$

$\implies\sf{\frac{144-169}{169}}$

$\implies\sf{\frac{-25}{169}\:\:[Answer]}$

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Formulas related to trigonometry :-

• sin²A + cos²A =1

• 1 + tan²A = sec²A

• 1 + cot²A = cosec²A

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