If tanA=n tanB and sinA=m sinB, prove that cos2A=n2−1m2−1.

We have to find cos2A in term of m and n. This means that the angle B is to be eliminated from the given relations.Now, tanA=ntanB⇒ tanB=n1tanA⇒ cotB=tanAnAnd sinA=msinB⇒sinB=m1sinA⇒ csc B=sinAmSubstituting the value of cotB and cscB in csc2B−cot2B=1, we getsin2Am2=tan2An2=1⇒ sin2Am2−sin2An2cos2A=1⇒ sin2Am2−n2cos2A=1⇒ sin2Am2−sin2Am2−n2cos2A=1⇒ sin2Am2−n2cos2A=1⇒ m2−n2cos2A=sin2A⇒ m2−n2cos2A=1−cos2A⇒ m2−1=n2cos2A−cos2A⇒ m2−1=(n2−1)cos2A⇒ n2−1m2−1=cos2A.

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If A=n B and A=m B, prove that ^2 A= {m^2-1}{n^2-1}.

Question

If $tan A = n tan B$ and $sin A = m sin B$ , prove that $cos^{2} A = \frac{m ^{2} - 1}{n ^{2} - 1}$ .

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Jack Discussed

tan A = n tan B

and

sin A = m sin B

, prove that

cos^{2} A = \frac{m ^{2} - 1}{n ^{2} - 1}

8 mins ago

Discuss this question LIVE

8 mins ago

Text solutionVerified

We have to find

cos^{2} A

in term of

m

and

n

. This means that the angle

B

is to be eliminated from the given relations.

Now,

tan A = n tan B

\Rightarrow tan B = \frac{1}{n} tan A \Rightarrow cot B = \frac{n}{tan A}

And

sin A = m sin B

\Rightarrow sin B = \frac{1}{m} sin A \Rightarrow csc B = \frac{m}{sin A}

Substituting the value of

cot B

and

csc B

csc^{2} B - cot^{2} B = 1

, we get

\frac{m ^{2}}{sin ^{2} A} = \frac{n ^{2}}{tan ^{2} A} = 1 \Rightarrow \frac{m ^{2}}{sin ^{2} A} - \frac{n ^{2} cos ^{2} A}{sin ^{2} A} = 1

\Rightarrow \frac{m ^{2} - n ^{2} cos ^{2} A}{sin ^{2} A} = 1 \Rightarrow \frac{m ^{2}}{sin ^{2} A} - \frac{m ^{2} - n ^{2} cos ^{2} A}{sin ^{2} A} = 1

\Rightarrow \frac{m ^{2} - n ^{2} cos ^{2} A}{sin ^{2} A} = 1 \Rightarrow m^{2} - n^{2} cos^{2} A = sin^{2} A

\Rightarrow m^{2} - n^{2} cos^{2} A = 1 - cos^{2} A \Rightarrow m^{2} - 1 = n^{2} cos^{2} A - cos^{2} A

\Rightarrow m^{2} - 1 = (n^{2} - 1) cos^{2} A \Rightarrow \frac{m ^{2} - 1}{n ^{2} - 1} = cos^{2} A

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Question Text	If $tan A = n tan B$ and $sin A = m sin B$ , prove that $cos^{2} A = \frac{m ^{2} - 1}{n ^{2} - 1}$ .
Answer Type	Text solution:1
Upvotes	150

If tanA=n tanB and sinA=m sinB, prove that cos2A=n2−1m2−1​.

Text solutionVerified

Practice questions from similar books

If $tan A = n tan B$ and $sin A = m sin B$ , prove that $cos^{2} A = \frac{m ^{2} - 1}{n ^{2} - 1}$ .