Toolbox:Rhombus is a parallelogram with all the four sides equal.If $\overrightarrow a.\overrightarr...

Toolbox:To prove that the parallelogram is a rectangle prove that the adjacent sides are $\perp$If $...

Toolbox:To prove that the $\perp$ bisectors are concurrent, prove that the third bisector passing th...

Toolbox:To prove altitudes are concurrent.draw a line through the intersection of two altitudes whic...

Toolbox: $\overrightarrow a.\overrightarrow b=|\overrightarrow a||\overrightarrow b|cos\th...

Toolbox: $\overrightarrow a.\overrightarrow b=|\overrightarrow a||\overrightarrow b|cos\th...

Toolbox: $|\overrightarrow a\times\overrightarrow b|=|\overrightarrow a||\overrightarrow b...

Toolbox:If $A(x_1,y_1,z_1)\:\:and\:\:B(x_2,y_2,z_2)$ are two points in space then $\overrightarrow {...

Toolbox: $|\overrightarrow a\times\overrightarrow b|=|\overrightarrow a||\overrightarrow b...

Toolbox:In a $\Delta\:ABC,\:\:\overrightarrow {AB}+\overrightarrow {BC}+\overrightarrow {CA}=\overri...

Toolbox: $n^{th}$ term, $t_n$ of an A.P. having ' A' as first term and 'd' as common diffe...

Toolbox:Triangular law of addition: In a $\Delta\:PQR,\:\overrightarrow {PQ}+\overrightarrow {QR}=\o...

Toolbox: If two vectors $\overrightarrow a\:and\:\overrightarrow b $ are perpendicular the...

Toolbox:Section formula: If C is mid point of AB then position vector of C,$\overrightarrow {OC}=\fr...

Toolbox:According to parallelogram law of addition, if $\large\overrightarrow a$ $and$ $\large\overr...

Toolbox: Area of parallelogram =$\large\frac{1}{2}|\overrightarrow a\times\overrightarrow ...

Toolbox: $\large|\overrightarrow a-\overrightarrow b|^2=(\overrightarrow a-\overrightarrow...

Toolbox:$\large|\overrightarrow a+\overrightarrow b|^2=(\overrightarrow a+\overrightarrow b).(\overr...

Toolbox: To prove angle between any two lines is rightangle, prove that the dot product of...

Toolbox:$\large|\overrightarrow a\times\overrightarrow b|=|\overrightarrow a||\overrightarrow b|sin\...

Toolbox:$\large|\overrightarrow a+\overrightarrow b|^2=|\overrightarrow a|^2+|\overrightarrow b|^2+|...

Toolbox: $\large\overrightarrow a \times\overrightarrow b=-\overrightarrow b\times\overrig...

Toolbox:If two vectors $\overrightarrow a\:and\:\overrightarrow b$ are parallel then $\overrightarro...

Toolbox: Area of $\Delta ABC=\frac{1}{2}|\overrightarrow {BC}\times\overrightarrow {BA}|$ ...

Toolbox:\(tan^{-1}x+cot^{-1}x=\frac{\pi}{2}\)\(cot\frac{\pi}{6}=\sqrt3\)Given equation can be writte...

Toolbox:\(tan(x-y)=\frac{tanx-tany}{1+tanx.tany}\)\(tan\frac{\pi}{4}=1\)Let \(x=tan\theta,\:\Rightar...

Toolbox: \( tan^{-1}\alpha-tan^{-1}\beta=tan^{-1}\large\frac{\alpha-\beta}{1+\alpha\beta}\:\:\alp...

Toolbox:Principal interval of sin is \([-\frac{\pi}{2},\frac{\pi}{2}]\)\(sin(\pi-\theta)=sin\theta\)...

First of all find the sum of the given two vectors \(\vec{a}=2i +4j-5k,\:and\:\vec{b}=i +2j+3k\) Th...

Ans: (A)  \( \frac{1}{\sqrt2}<x\leq\:1\)   Because when \(x\in\:\big(\frac{1}{\sqrt2}...

Toolbox: \( 1-cos\theta=2sin^2\large\frac{\theta}{2}\) \( sin\theta=2sin\large\frac{...

Toolbox: \( 2tan^{-1}x=tan^{-1}\large\frac{2x}{1-x^2} for\:|x|<1\) \( tan^{-1}x+t...

Ans: A \[Sin(\frac{\pi}{3}-\frac{\pi}{6})=Sin\frac{\pi}{6}=\frac{1}{2}\]

\[Ans:\:\frac{2\pi}{3}+\frac{\pi}{3}=\pi\]

Ans: \[\frac{\pi}{6}\]

Ans: \[Cos^{-1}Cos(\pi+\frac{\pi}{6})=Cos^{-1}cos(\pi-\frac{\pi}{6})=\frac{\pi}{6}\]

Ans: \[Sin^{-1}sin(\pi-\frac{2\pi}{5})=Sin^{-1}sin\frac{2\pi}{5}=\frac{2\pi}{5}\]

Ans: \[\frac{\pi}{4}+(\pi-\frac{\pi}{3})-\frac{\pi}{6}=\frac{3\pi}{4}\]

Ans: \[\frac{x+y}{1-xy}\]

Ans: x=4,because \[tan^{-1}x+cot^{-1}x=\frac{\pi}{2}\]

Ans:\[\frac{\pi}{3}+\frac{2\pi}{6}=\frac{2\pi}{3}\]

Ans: \[\frac{2\pi}{3}\]

Ans: \[Tan^{-1}tan(\pi+\frac{\pi}{6})=Tan^{-1}tan\frac{\pi}{6}=\frac{\pi}{6}\]

Ans:  By taking x= SinØ we can prove the result.

Ans:  \[\frac{1}{3}\]

Tool Box: \[Take\:x=CosA\:then\:Cos^{-1}x=A\] \[L.H.S.\:becomes.Cos^{-1}Cos3A=3A=3Cos^{-1}x\]

Ans: \[\frac{\pi}{3}-\pi+\frac{\pi}{3}=\frac{2\pi}{3}\]

Ans: \[sin\frac{\pi}{2}=1\]

Tool box: \[Tan^{-1}x+Tan^{-1}y=Tan^{-1}\:\frac{x+y}{1-xy}\] using the forfula above prove the res...

Tool box: \[Sin^{-1}x+Cos^{-1}x=\frac{\pi}{2}\] \[Sin\frac{\pi}{2}=1\] Ans: \[x=\frac{1}{5}\]