Question

if sintheta+costheta=√2sin(90°-theta) show that cot theta=(√2+1)​

Answer 1

$\underline{\textsf{Given:}}$

$\mathsf{sin\theta+cos\theta=\sqrt{2}\,sin(90^{\circ}-\theta)}$

$\underline{\textsf{To prove:}}$

$\mathsf{cot\theta=\sqrt{2}+1}$

$\underline{\textsf{Solution:}}$

$\textsf{Consider,}$

$\mathsf{sin\theta+cos\theta=\sqrt{2}\,sin(90^{\circ}-\theta)}$

$\mathsf{sin\theta+cos\theta=\sqrt{2}\,cos\theta}$

$\textsf{Divide bothsides of the equation by}$

$\mathsf{cos\theta}$

$\mathsf{tan\theta+1=\sqrt{2}}$

$\mathsf{tan\theta=\sqrt{2}-1}$

$\textsf{Taking reciprocals we get}$

$\mathsf{cot\theta=\dfrac{1}{\sqrt{2}-1}}$

$\textsf{For rationalizing the denominator}$

$\mathsf{cot\theta=\dfrac{1}{\sqrt{2}-1}{\times}\dfrac{\sqrt{2}+1}{\sqrt{2}+1}}$

$\mathsf{cot\theta=\dfrac{\sqrt{2}+1}{\sqrt{2}^2-1^2}}$

$\mathsf{cot\theta=\dfrac{\sqrt{2}+1}{2-1}}$

$\mathsf{cot\theta=\dfrac{\sqrt{2}+1}{1}}$

$\implies\boxed{\mathsf{cot\theta=\sqrt{2}+1}}$

Answer 2

Given:

sintheta+costheta=√2sin(90°-theta)

To find:

if sintheta+costheta=√2sin(90°-theta) show that cot theta=(√2+1)​

Solution:

From given, we have,

sin theta + cos theta = √2 sin(90° - theta)

here, we use the trigonometric property,

sin(90° - theta) = cos theta

sin theta + cos theta = √2 cos theta

sin theta  = √2 cos theta - cos theta

sin theta  = (√2 - 1) cos theta

here, we use the trigonometric ratios

cot theta = cos theta / sin theta

sin theta  = (√2 - 1) cos theta

divide the above equation by sin theta,

sin theta / sin theta  = (√2 - 1) cos theta / sin theta

1 = (√2 - 1) cot theta

cot theta = 1/(√2 - 1)

use the rationalizing property, (1/(√2 - 1) = √2 + 1)

cot theta = √2 + 1

Hence it is shown that, cot theta = √2 + 1