Show thet: 3( - )+6( + )^2+4( ^6x+ ^6x)=13 - Vidyadip

Question

Show thet:
$3(\sin\text{x}-\cos\text{x})+6(\sin\text{x}+\cos)^2+4(\sin^6\text{x}+\cos^6\text{x})=13$

✓

Answer

$\text{LHS}=3(\sin\text{x}-\cos\text{x})+6(\sin\text{x}+\cos)^2+4(\sin^6\text{x}+\cos^6\text{x})$
$=3[\sin^4\text{x}-4\sin^3\text{x}\cos\text{x}+6\sin^2\text{x}\cos^2\text{x}-4\sin\text{x}\cos^3\text{x}+\cos^4\text{x}]$
$+6[\sin^2\text{x}+2\sin\text{x}\cos\text{x}+\cos^2\text{x}]+4(\sin^6\text{x}+\cos^6\text{x})$
$\big[\because(\text{a}-\text{b})^4=\text{a}^4-4\text{a}^3\text{b}+6\text{a}^2\text{b}^2-4\text{ab}^3+\text{b}^4$ by binomial expainsion$\big]$
$=3\big[\sin^4\text{x}+\cos^4\text{x}-4\sin\text{x}\cos\text{x}(\sin^2\text{x}+\cos^2\text{x})+6\sin^2\text{x}\cos^2\text{x}\big]$
$+6[1+2\sin\text{x}\cos\text{x}]+4\big[(\cos^2\text{x}+\sin^2\text{x})(\cos^4\text{x}-\cos^2\text{x}+\sin^4\text{x})\big]$
$\big[\because\text{a}^3+\text{b}^3=(\text{a+b})(\text{a}^2-\text{ab+b}^2)$
$=7[\sin^4\text{x}+\cos^4\text{x}]+18\sin^2\text{x}\cos^4\text{x}-4\sin^2\text{x}\cos^2\text{x}+6$
$=7[\sin^4\text{x}+\cos^4\text{x}2\sin^2\cos^2]+6$
$=7[\sin^2\text{x}+\cos^2\text{x}+2\sin^2\text{x}\cos^2\text{x}]+6$
$=7+6$
$=13=\text{RHS}$

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 "5 Marks Questions" from Trigonometric Functions→Trigonometric Functions — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Find the distance of the line 2x + y = 3 from the point (-1, -3) in the direction of the line whose slope is 1.

If A = {1, 2, 3}, B = {4}, C = {5}, then verify that:
$\text{A}\times(\text{B}-\text{C})=(\text{A}\times\text{B})-(\text{A}\times\text{C})$

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{27}}\frac{\Big(\text{x}^\frac{1}{3}+3\Big)\Big(\text{x}^{\frac{1}{3}}-3\Big)}{\text{x}-27}$

Prove the following statement by principle of mathematical induction:
$1 + 5 + 9 + ...... + (4n - 3) = n(2n - 1),$ for all natural numbers $n.$

$5^{2n} - 1$ is divisible by 24 for all $\text{n}\in\text{N}.$

If a denotes the number of permutations of (x + 2) things taken all at a time, b the number of permutations of x things taken 11 at a time and c the number of permutations of x − 11 things taken all at a time such that a = 182 bc, find the value of x.

Find the sum of the following series to n terms:
$1 + (1 + 2) + (1 + 2 + 3) + (1 + 2 + 3 + 4) + ...$

One of the four persons John, Rita, Aslam or Gurpreet will be promoted next month. Consequently the sample space consists of four elementary outcomes $S = \{$John promoted, Rita promoted, Aslam promoted, Gurpreet promoted$\}$ You are told that the chances of John’s promotion is same as that of Gurpreet, Rita’s chances of promotion are twice as likely as Johns. Aslam’s chances are four times that of John:

Determine $P($John promoted$)$

$P($Rita promoted$)$
$P($Aslam promoted$)$
$P($Gurpreet promoted$)$

If $A = \{$John promoted or Gurpreet promoted$\}$, find $P(A).$

Evaluate the following:
$\text{x}^4-4\text{x}^3+4\text{x}^2+8\text{x}+44,$ when $\text{x}=3+2\text{i}$

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{3\sin\text{x}-\sin3\text{x}}{\text{x}^3}$