Question

in the given figure , angle p = theta and angle r = phie. find √x+1cotphie

Answer 1

Please see the attachment

Answer 2

Answer:  The answer is $\dfrac{x}{2}.$

Step-by-step explanation:  We are given a right-angled triangle PQR, where

$QR=x,~~PR=x+2,~~\angle RPQ=\theta,~~\angle PRQ=\phi.$

Let us consider that PQ = y as shown in the attached figure.

Now, using Pythagoras Theorem, we have from Δ PQR that

$PR^2=PQ^2+QR^2\\\\\Rightarrow (x+2)^2=y^2+x^2\\\\\Rightarrow y^2=(x+2)^2-x^2\\\\\Rightarrow y^2=x^2+4x+4-x^2\\\\\Rightarrow y^2=4(x+1)\\\\\Rightarrow y=2\sqrt{x+1}.$

Also, using trigonometric ratios, we can write

$\cot \phi=\dfrac{QR}{PQ}\\\\\\\Rightarrow \cot \phi=\dfrac{x}{y}\\\\\\\Rightarrow \cot \phi=\dfrac{x}{2\sqrt{x+1}}\\\\\\\Rightarrow \sqrt{x+1}\cot \phi=\dfrac{x}{2}.$

Thus, the required value is $\dfrac{x}{2}.$