Prove that ^ -1 4 5 + ^ -1 12 13 = ^ -1 33 65 - Vidyadip

Question

Prove that $\cos^{-1}\frac{4}{5}+\cos^{-1}\frac{12}{13}=\cos^{-1}\frac{33}{65}$

✓

Answer

$\text{L.H.S}=\cos^{-1}\frac{4}{5}+\cos^{-1}\frac{12}{13}$
$ =\cos^{-1}\Bigg[\frac{4}{5}\times\frac{12}{13}-\sqrt{1-\Big(\frac{4}{5}\Big)^2}\sqrt{1-\Big(\frac{12}{13}\Big)^2}\Bigg)$
$\Big[\because\ \cos^{-1}\text{x}+\cos^{-1}\text{y}=\cos^{-1}\Big(\text{xy}-\sqrt{1-\text{x}^2}\sqrt{1-\text{y}^2}\Big)\Big]$
$ =\cos^{-1}\Big[\frac{48}{65}-\frac{3}{5}\times\frac{5}{12}\Big]$
$=\cos^{-1}\Big(\frac{48-15}{65}\Big)$
$=\cos^{-1}\frac{33}{65}=\text{R.H.S}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from Inverse Trigonometric Functions→Inverse Trigonometric Functions — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Write the set of values of a for which $\text{f}(\text{x})=\cos\text{x}+\text{a}^2\text{x}+\text{b}$ is strictly increasing on R.

A line passes throuth the point with position vector $2\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}}$ and is in the direction of $3\hat{\text{i}}+4\hat{\text{j}}-5\hat{\text{k}}.$ Find equations of the line in vector and cartesian form.

Dot product of a vector with vectore $\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}},2\hat{\text{i}}+\hat{\text{j}}-3\hat{\text{k}}$ and $\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ are respectively 4, 0 and 2. Find the vector.

Three relation $R_{4 }$ is defined in set $A = \{a, b, c\}$ as follows: $R_4 = \{(a, b), (b, c), (c, a)\}$
Find whether or not the relation $R_{4 }$ on $A$ is:

Reflexive.
Symmetric.
Transitive.

Determine P(E|F) in Exercises.
Two coins are tossed once, where
E : no tail appears, F : no head appears.

If the mean and variance of a binomial distribution are respectively 9 and 6, find the distribution.

Evaluate the following definite integrals:
$\int_{0}^\limits{\pi}\frac{1}{1+\sin\text{x}}\text{ dx}$

Let $\vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}$ and $\vec{\text{c}}=\text{c}_1\hat{\text{i}}+\text{c}_2\hat{\text{j}}+\text{c}_3\hat{\text{k}}.$ Then, If $C_2 = -1$ and $C_3 = 1,$ show that no value of $C_1$ can make $\vec{\text{a}},\vec{\text{b}}$ and $\vec{\text{c}}$ coplanar.

A pair of dice is thrown 6 times. If getting a total of 9 is considered a success, what is the probability of at least 5 successes?

Find the probability distribution of the number of heads, when three coins are tossed.