If sintheta + costheta = root , then prove that tantheta + costheta =1
Answers
Answer:
tanθ+cotθ=1 , If the value of \bold{\sin \theta+\cos \theta=\sqrt{3}}sinθ+cosθ=
3
Given:
\sin \theta+\cos \theta=\sqrt{3}sinθ+cosθ=
3
To Prove:
\tan \theta+\cot \theta=1tanθ+cotθ=1
Proof:
\sin \theta+\cos \theta=\sqrt{3}sinθ+cosθ=
3
Squaring of both sides, we get:
(\sin \theta+\cos \theta)^{2}=(\sqrt{3})^{2}(sinθ+cosθ)
2
=(
3
)
2
Using the formula (a+b)^{2}=a^{2}+b^{2}+2 a b(a+b)
2
=a
2
+b
2
+2ab
Applying formula in (\sin \theta+\cos \theta)^{2},(sinθ+cosθ)
2
,
\begin{lgathered}\begin{array}{l}{\sin ^{2} \theta+\cos ^{2} \theta+2 \sin \theta \cos \theta=3} \\ {\because \sin ^{2} \theta+\cos ^{2} \theta=1}\end{array}\end{lgathered}
sin
2
θ+cos
2
θ+2sinθcosθ=3
∵sin
2
θ+cos
2
θ=1
1+2sinθcosθ=3
2sinθcosθ=2
sinθcosθ=1 ______(1)
The value of the \sin \theta+\cos \theta=\sqrt{3}sinθ+cosθ=
3
is sinθcosθ=1
To prove:
tanθ+cotθ=1
L.H.S
tanθ+cotθ
Transforming the identity of tanθ ; cotθ into \frac{\sin \theta}{\cos \theta}
cosθ
sinθ
; \frac{\cos \theta}{\sin \theta}
sinθ
cosθ
\frac{\sin \theta}{\cos \theta}+\frac{\cos \theta}{\sin \theta}
cosθ
sinθ
+
sinθ
cosθ
\frac{\sin ^{2} \theta+\cos ^{2} \theta}{\sin \theta \cos \theta}
sinθcosθ
sin
2
θ+cos
2
θ
Substituting equation (1) we get
\begin{lgathered}\begin{array}{l}{\frac{\sin ^{2} \theta+\cos ^{2} \theta}{1}} \\ {\because \sin ^{2} \theta+\cos ^{2} \theta=1}\end{array}\end{lgathered}
1
sin
2
θ+cos
2
θ
∵sin
2
θ+cos
2
θ=1
tanθ+cotθ=1=R.H.S
∴L.H.S=R.H.S
Hence proved
∴If \bold{\sin \theta+\cos \theta=\sqrt{3}}sinθ+cosθ=
3
then \bold{\tan \theta+\cot \theta=1}tanθ+cotθ=1