Question

solve the following with solutions with clear explanation

Answer 1

Answer:

sin+cos=p

tan+cot=q

sin/cos+cos/sin=q

sin2+cos2/sin.cos(sin2+cos2+2sin.cos-1)

1/sin.cos(2sin.cos)

=2

Answer 2

${\bold{\underline{\underline{Answer:}}}}$

${\bold{\therefore q({p}^{2}-1)=2 }}$

${\bold{\therefore \sqrt{1-sin\:100\degree}\times sec\:100\degree=1 }}$

${\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\bold{For \: first \: question : } \\ \\ \underline \bold{Given : } \\ \implies sin \theta + cos \theta = p \\ \\ \implies \tan \theta + \cot \theta = q \\ \\ \underline \bold{To \: find : } \\ \implies q( {p}^{2} - 1) =?$

$\implies \tan \theta + \cot \theta = q \\ \\ \implies \frac{sin \theta}{cos \theta} + \frac{cos \theta}{sin \theta} = q \\ \\ \implies \frac{ {sin}^{2} \theta + {cos}^{2} \theta }{sin \theta\times cos \theta} = q \\ \\ \implies q = \frac{1}{ sin \theta \times cos \theta} \\ \\ \bold{For \: finding \: values : } \\ \implies q( {p}^{2} - 1) \\ \\ \implies \frac{1}{sin \theta \times cos \theta}( ( {sin \theta + cos \theta)}^{2} - 1) \\ \\ \implies \frac{1}{sin \theta \times cos \theta} ({sin}^{2} \theta + {cos}^{2} \theta + 2 \: sin \theta \times cos \theta - 1 )\\ \\ \implies \frac{1}{sin \theta \times cos \theta} (1 + 2 \: sin \theta \times cos \theta - 1) \\ \\ \implies \frac{1}{sin \theta \times cos \theta} \times 2 \: sin \theta \times cos \theta \\ \\ \implies \bold{2}$

$\bold{For \: second \: question : } \\ \implies \sqrt{1 - {sin}^{2}100 \degree } \times sec \: 100 \degree \\ \\ \bold{ cos \: theta = \sqrt{1 - sin ^{2} \theta} } \\ \\ \implies cos \: 100 \degree \times sec \: 100 \degree \\ \\ \implies \frac{ 1 }{ \cancel{sec \:100 \degree}} \times \cancel{sec \:100 \degree} \\ \\ \bold{\implies 1}$